BF 


UC-NRLF 


of  Cbicago 


HOW  NUMERALS  ARE  READ 

AN  EXPERIMENTAL  STUDY  OF  THE  READING  OF 

ISOLATED  NUMERALS  AND  NUMERALS 

IN  ARITHMETIC  PROBLEMS 


A  DISSERTATION 

SUBMITTED  TO  THE  FACULTY 

OF  THE  GRADUATE  SCHOOL  OF  ARTS  AND  LITERATURE 

IN  CANDIDACY  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 

DEPARTMENT  OF  EDUCATION 


BY 


PAUL  WASHINGTON  TERRY 


Private  Edition,  Distributed  By 

THE  UNIVERSITY  OF  CHICAGO  LIBRARIES 

CHICAGO,  ILLINOIS 


Reprinted  from 
SUPPLEMENTARY  EDUCATIONAL  MONOGRAPHS,  No.  18,  June,  1922 


EDUCATION  DEP 


Ube  ZHniversfts  of  Gbtca'aicL      v -X  :*'•: !/ 


HOW  NUMERALS  ARE  READ 

AN  EXPERIMENTAL  STUDY  OF  THE  READING  OF 

ISOLATED  NUMERALS  AND  NUMERALS 

IN  ARITHMETIC  PROBLEMS 


A  DISSERTATION 

SUBMITTED  TO  THE  FACULTY 

OF  THE  GRADUATE  SCHOOL  OF  ARTS  AND  LITERATURE 

IN  CANDIDACY  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 

DEPARTMENT  OF  EDUCATION 


BY 

PAUL  WASHINGTON  TERRY 


Private  Edition,  Distributed  By 

THE  UNIVERSITY  OF  CHICAGO  LIBRARIES 

CHICAGO,  ILLINOIS 


Reprinted  from 
SUPPLEMENTARY  EDUCATIONAL  MONOGRAPHS,  No.  18,  June,  1922 


COPYRIGHT  1922  BY 
THE  UNIVERSITY  OF  CHICAGO 


All  Rights  Reserved 


Published  June  1922 

EDUCATION  DEPfT 


TABLE  OF  CONTENTS 

PAGE 

LIST  OF  PLATES ix 

LIST  OF  TABLES .  xi 

LIST  OF  SELECTIONS xiii 

CHAPTER  I.  INTRODUCTION '    .  i 

PART  I.     PRELIMINARY  STUDIES  OF  THE  READING  OF 

NUMERALS— BY  INTROSPECTIVE  METHODS 
CHAPTER      II.  NUMERALS  IN  ARITHMETICAL  PROBLEMS — FIRST  PRE- 
LIMINARY STUDY 2 

1.  Description  of  the  Study.        ...       .       .       .  2 

2.  First  Reading  and  Re-reading  Distinguished       .       .  3 

3.  Partial  First  Reading  and  Whole  First  Reading  of 
Numerals    .        .        .        . .3 

4.  Individual  Subjects  Classified  as  Partial  First  Readers 

and  as  Whole  First  Readers 6 

5.  Re-reading  of  the  Several  Numerals      .        .       .       .  7 

6.  Re-reading  by  Individual  Subjects        .       .       .       .  9 

7.  Summary 10 

CHAPTER    III.  RANGE  OF  CORRECT  RECALL  OF  NUMERALS  AFTER  THE 

FIRST  READING — SECOND  PRELIMINARY  STUDY       .       .  12 

1.  Description  of  the  Study 12 

2.  Range  of  Recall  of  the  Several  Numerals     .        .        .  14 

3.  Range  of  Recall — by  the  Several  Subjects  16 

4.  Further  Evidence  as  to  the  Purpose  of  First  Reading  .  17 

5.  Items  of  Recall  Not  Included  in  the  Classifications  .  17 

6.  Summary  of  Conclusions 17 

CHAPTER     IV.  ANALYSIS  OF  THE  RE-READING  OF  NUMERALS  IN  ARITH- 
METICAL PROBLEMS — THIRD  PRELIMINARY  STUDY  .       .  19 

1.  Description  of  the  Study 19 

2.  Objects  and  Nature  of  the  Re-readings        .       .       .  22 

3.  Duration  of  Re-readings 22 

4.  Summary    .               23 

CHAPTER      V.  READING    NUMERALS    IN     COLUMNS — FOURTH     PRE- 
LIMINARY STUDY 24 

1.  Description  of  the  Study 24 

2.  General  Description  of  the  Three  Main  Groups  Used  25 

3.  Main-Group  Patterns  for  Numerals  of  Like  Length  .  27 


VI  TABLE  OF  CONTENTS 

PAGE 

4.  Variations  in  Numerical  Language        .       .        .       .  29 

5.  Influence  of  Punctuation  on  the  Grouping  of  Digits 

of  Longer  Numerals 31 

6.  Persistence    of    Patterns    from    the    First    Reading 
through  a  Second  Reading 33 

7.  Summary  of  Conclusions 33 

PART  II.    STUDIES  OF  THE  READING  OF  NUMERALS— 
BY  USE  OF  PHOTOGRAPHIC  APPARATUS 

CHAPTER     VI.  DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES       .       .  35 

1.  Apparatus  Described 35 

2.  Three  Types  of  Reading-Materials  Used      .       .       .  35 

3.  Instructions  to  Subjects  and  Description  of  Subjects  38 

4.  Procedure  on  the  Part  of  the  Observer.       ...  39 

5.  Guide  for  Reading  the  Plates. 39 

6.  Plates  I-XV 41-54 

CHAPTER   VII.  FIRST  READING  OF  NUMERALS  IN  PROBLEMS  ...  56 

1.  Introduction 56 

2.  Comparison  between  the  Reading  of  Numerals  and 
Words  in  Problems 56 

3.  Partial  and  Whole  First  Reading  of  Numerals    .       .  59 

4.  The  Several  Subjects  as  Partial  and  Whole  First 
Readers 64 

5.  Relative  Value  of  Partial  and  of  Whole  First  Reading  64 

6.  Development  of  the  Method  of  Partial  First  Reading  66 

7.  Summary  of  Conclusions         .              •.       .       .       .  67 

CHAPTER  VIII.  RE-READING  AND  COMPUTATION 69 

1.  Two  Types  of  Re-reading  of  Numerals.       ...  69 

2.  Methods  and  Procedures  Used   in  the  Process  of 
Computation 72 

3.  Summary 76 

CHAPTER    IX.  THE  READING  OF  ISOLATED  NUMERALS  IN  LINES   .       .  78 

1.  Introduction 78 

Previous  Investigations  by  Gray  and  Dearborn         .  78 

Descriptions  of  Plates      . 79 

Plates  XVI-XXV     .......         80-84 

2.  Two  Types  of  Pauses 85 

3.  Differences  in  the  Readings  of  Numerals  of  Different 
Lengths 86 

4.  The  Special  Numerals      .       .       .       .  .       .90 

5.  Two  Methods  of  Attack  in  Reading  Numerals   .       .  91 

6.  Summary 92 


TABLE  OF  CONTENTS  vil 

PAGE 

CHAPTER      X.  COMPARISONS  or  RATES  OF  READING 94 

1.  Comparison  of  the  Subjects  of  the  Present  Investiga- 
tion with  the  Subjects  of  Schmidt's  Investigation  in 
Respect  to  Rates  of  Reading 94 

2.  Comparisons  of  Rates  of  Reading  the  Three  Types  of 
Reading-Materials 96 

3.  Summary 97 

CHAPTER    XI.  PRACTICAL  APPLICATION  TO  CLASSROOM  TEACHING  .       .  98 

1.  The  Question  of  Reading  in  Arithmetic       ...  98 

2.  Preliminary  Analytical  Reading  of  Problems       .       .  99 

3.  Application  of  Partial  Reading 100 

4.  Application  of  Re-reading 102 

5.  Miscellaneous  Applications 104 

INDEX  .                                                                             ....  107 


LIST  OF  PLATES 

PLATE  AGE 

I.  First  Reading  of  Problem  4  by  Subject  B  and  His  Pro- 
cedure in  Solving  the  Problem 41 

II.  First  Reading  of  Problem  4  by  Subject  G  and  Re-reading 

the  Numeral  1000 42 

III.  First  Reading  of  Problem  i  by  Subject  W  and  Multiplica- 

tion Direct  from  the  Problem  Card  with  One  Numeral 
Used  as  the  "Base  of  Operations" 43 

IV.  First  Reading  of  Problem  3  by  Subject  Hb  and  Re- 

reading the  Numerals  for  Copying  .        .        .        .        .       44 
V.  First  Reading  of  Problem  2  by  Subject  Hb,  Re-reading 
the   First   Numeral,   and   Subsequent   Re-reading   of 

Both  Numerals  for  Copying 45 

VI.  First  Reading  of  Problem  5  by  Subject  M  and  Re-reading 

and  Copying  the  Two  Numerals      .     •  .        .        .        .46 
VII.  First  Reading  of  Problem  i  by  Subject  Hb  and  Re- 
reading the  First  Numeral  for  Copying  ....       47 
VIII.  First  Reading  of  Problem  i  by  Subject  B  and  the  Process 
of  Computation  with  the  First  Numeral  as  the  "Base 

of  Operations" 48 

IX.  First  Reading  of  Problem  2  by  Subject  G  and  the  Pro- 
cess of  Adding  the  Two  Numerals 49 

X.  First  Reading  of  Problem  i  by  Subject  G  and  the  Process 

of  Computation 50 

XI.  First  Reading  of  Problem  i  by  Subject  M,  Re-reading 

Words,  and  the  Process  of  Computation        ...       50 
XII.  First  Reading  of  Problem  3  by  Subject  G  and  the  Process 

of  Computation 51 

XIII.  First  Reading  of  Problem  5  by  Subject  H  and  the  Process 

of  Computation 52 

XIV.  First  Reading  of  Problem  3  by  Subject  W  with  Partial 

Reading  of  Numerals 53 

XV.  First  Reading  of  Problem  5  by  Subject  G  and  the  Process 

of  Computation 54 

XVI-XVII.  Reading  of  Isolated  Numerals  by  Subject  G      ...       80 

XVIII-XIX.  Reading  of  Isolated  Numerals  by  Subject  H     .       .       .81 

XX-XXI.  Reading  of  Isolated  Numerals  by  Subject  M     .       .       .82 

XXII-XXIII.  Reading  of  Isolated  Numerals  by  Subject  B      .       .       .83 

XXIV-XXV.  Reading  of  Isolated  Numerals  by  Subject  W     .       .       .84 

ix 


LIST  OF  TABLES 

TABLE  PAGE 

I.  Partial  First  Readings  and  Whole  First  Readings  by  Ten 

Subjects  in  Seven  Problems 4 

II.  Ranks  of  Numerals  according  to  Percentages  of  Partial  First 

Readings 5 

III.  Subjects  Arranged  according  to  Number  of  Partial  and  Whole 

First  Readings 7 

IV.  Number  of  Re-readings 7 

V.  Ranks  of  Numerals  according  to  Percentages  of  Re-readings  .         8 

VI.  Number  of  Re-readings  by  Various  Subjects      ....         9 

VII.  Range  of  Correct  Recall  of  Numerals  from  First  Reading  of 

Problems 14 

VIII.  Varying  Ranges  of  Correct  Recall  of  Three-  to  Seven-Digit 

Numerals  by  the  Several  Subjects    .       .       .       .       .       .       16 

IX.  Numerals  and  Words  Read  at  Each  Re-reading  Together  with 

Number  of  Seconds  Required 21 

X.  Number  of   Seconds  Required   for  First  Reading  and  for 

Re-reading  of  Problems 23 

XI.  Main  Group  Patterns  Used  in  Reading  One-  to  Seven-Digit 

Numerals  in  Columns 26 

XII.  Number  of  One-,  Two-,  and  Three-Digit  Groups  Used  in 

Reading  Numerals  of  the  Several  Digit-Lengths  in  Columns      28 

XIII.  Number  of  Simple  (3)  and  of  Complex  (1-2)  Three-Digit 

Groups   Used   in   Reading   Five-,    Six-,   and   Seven-Digit 
Numerals  in  Columns 28 

XIV.  Numerical  Language  Patterns  Used  by  Subject  G  in  Reading 

Numerals  in  Columns 30 

XV.  Effect  of  Punctuation  on  the  Number  of  Two-  and  Three- 
Digit  Groups  Used  in  Reading  Five-,  Six-,  and  Seven-Digit 
Numerals  in  Columns 32 

XVI.  Description  of  the  Five  Problems  Read  before  the  Photo- 
graphic Apparatus 36 

XVII.  Average  Number  of  Digits  Included  in  a  Pause  on  Numerals 
Contrasted  with  Average  Number  of  Letters  Included  in 
a  Pause  on  Words  during  First  Reading  .  .  .  .  .  57 

xi 


xii  LIST  OF  TABLES 

TABLE  PAGE 

XVIII.  Average .  Duration  of  Pauses  in  Fiftieths  of  a  Second  on 
Numerals  Contrasted  with  Average  Duration  of  Pauses  on 
Words  during  First  Reading 58 

XIX.  Percentage  of  Regressive  Pauses  on  Numerals  Contrasted 
with  Percentage  of  Regressive  Pauses  on  Words  during 
First  Reading 58 

XX.  Duration  in  Fiftieths  of  a  Second,  and  Serial  Order  of  the 
Several  Pauses  Used  in  Whole  and  Partial  First  Readings  of 
Numerals 60 

XXI.  First  Reading  of  Numerals  in  Problems  Contrasted  with  the 

Reading  of  Isolated  Numerals  of  Corresponding  Lengths    .       62 

XXII.  Reading  of  Numerals  in  Problems  by  Partial  First  Readers 
Contrasted  with  Reading  of  Numerals  in  Problems  by 
Whole  First  Readers 64 

XXIII.  Reading  of  Words  in   Problems  by   Partial   First  Readers 

Contrasted  with  Reading  of  Words  in  Problems  by  Whole 
First  Readers 65 

XXIV.  Type  of  Re-reading  Given  to  Numerals,  or  to  Numerals  and 

Words,  before  Beginning  of  Computation       ....       70 

XXV.  Two  Methods  of  Proceeding  with  Numerals  for  Purposes  of 

Computation  after  the  First  Reading 72 

XXVI.  Analysis  of  the  Process  of  Computation  in  Which  One  Numeral 

Is  Used  as  the  " Base  of  Operations" 75 

XXVII.  Average  Number  of  Pauses  and  Average  Reading-Time  per 

Numeral  for  Isolated  Numerals 87 

XXVIII.  Average  Pause-Duration  of  Isolated  Numerals  of  the  Several 

Digit-Lengths 88 

XXIX.  Number  of  Isolated  Numerals  of  the  Several  Digit-Lengths 
Which  Were  Read  with  Various  Numbers  of  Pauses  per 
Numeral  ...  . 89 

XXX.  Reading  of  Special  Numerals   Compared  with  Reading  of 

Other  Isolated  Numerals  of  Corresponding  Digit-Lengths  .       90 

XXXI.  Speed  and  the  Two  Methods  of  Attack  Used  in  Reading 

Isolated  Numerals 91 

XXXII.  Subjects  of  the  Present  Investigation  Compared  with  Those 

of  Schmidt's  Investigation  in  Respect  to  Speed  of  Reading  .       95 

XXXIII.  Comparative  Data  from  Readings  of  Five  Problems,  Ordinary 

Prose,  and  Isolated  Numerals 96 


LIST  OF  SELECTIONS 

SELECTION  PAGE 

1.  Five  Problems  Read  before  Photographic  Apparatus     .  .       .       .       36 

2.  Isolated  Numerals  Read  before  Photographic  Apparatus  .       .       .       37 

3.  Ordinary  Prose  Read  before  Photographic  Apparatus    .  ...       38 


xiii 


CHAPTER  I 

INTRODUCTION 

Numerous  investigations  in  the  psychology  of  reading  and  in  the 
measurement  of  reading  ability  have  developed  a  valuable  body  of 
scientific  information  about  the  methods  of  reading  words  and  sentences 
of  the  ordinary  kind,  but  the  reading  of  numerals  has  had  only 
occasional  attention  in  these  studies  and  that  merely  in  an  incidental 
way.  Such  work  on  reading  as  has  been  done  in  the  field  of  arithmetic 
has  concerned  itself  with  isolated  numerals  rather  than  with  numerals 
set  in  sentences  or  problems. 

The  present  investigation  is  concerned  with  the  reading  of  numerals 
both  in  separate  lines  and  in  the  context  of  arithmetical  problems.  The 
first  part  of  this  report  describes  a  series  of  studies,  based  on  introspective 
observations,  of  some  of  the  relatively  definite  and  highly  developed 
habits  of  graduate  students  in  reading  numerals  both  isolated  and  in 
problems.  The  second  part  of  the  report  deals  with  this  same  class  of 
readers  and  with  the  same  kinds  of  reading  materials,  but  employs 
objective  methods. 

For  the  first  part  of  the  report  the  data  were  obtained  by  recording 
introspective  observations  which  were  made  by  the  subjects  after  they 
had  read  a  set  of  arithmetical  problems  in  which  numerals  occurred. 
The  introspections  were  supplemented  by  directly  observing  and  report- 
ing the  results  of  the  reading  of  isolated  numerals.  The  information 
secured  in  this  preliminary  work  serves  as  a  basis  for  the  interpretation 
of  the  data  obtained  in  the  second  part  of  the  investigation  in  which 
photographic  records  of  eye-movements  were  secured.  The  whole 
investigation  is  only  an  introduction  to  the  study  of  the  methods 
employed  by  children  in  their  gradual  acquisition  of  the  power  of  reading 
numerals.  This  large  genetic  study  was  the  original  aim  of  the  present 
investigation.  The  intricacies  of  the  problem  turned  what  was  originally 
thought  of  as  an  introductory  investigation  into  an  elaborate  detailed 
study.  Yet  educational  implications  are  present  even  in  this  preliminary 
work.  Through  the  study  of  adults,  a  body  of  facts  has  been  discovered 
which  throws  light  on  methods  of  reading  problems  in  arithmetic  to 
which  children  must  ultimately  attain,  whatever  be  the  initial  habits 
through  which  they  pass  in  the  course  of  their  development. 


PART  I.     PRELIMINARY  STUDIES  OF  THE  READING  OF 
NUMERALS— BY  INTROSPECTIVE  METHODS 

CHAPTER  II 

NUMERALS  IN  ARITHMETICAL  PROBLEMS— FIRST 
PRELIMINARY  STUDY 

I.      DESCRIPTION   OF   THE   STUDY 

For  the  first  preliminary  study  seven  simple  arithmetical  problems 
were  used.  These  were  so  formulated  that  each  included  a  set  of  from 
one  to  four  numerals.  The  problems  were  so  made  up  that  while  the 
numerals  in  each  one  were  similar,  those  which  Were  used  in  the  different 
problems  exhibited  variations  in  length.  The  numerals  in  problems 
i,  3,  5,  and  6  are  'two  in  number  in  each  case,  but  vary  in  digit-length 
from  one  to  seven  digits.  Problem  2  includes  a  set  of  four  numerals, 
each  numeral  being  made  up  of  from  one  to  two  digits;  Problem  4  has 
four  numerals  made  up  of  from  three  to  four  digits;  and  Problem  / 
uses  a  familiar  date  and  two  numerals  of  exceptional  character,  namely 
100  and  1000. 

The  problems  which  were  used  in  the  first  preliminary  study  are  as 
follows : 

1.  At  65  cents  a  dozen,  what  will  8  dozen  eggs  cost  ? 

2.  A  man  buys  5  tons  of  coal  at  9  dollars  a  ton,  and  3  cords  of  wood  at 
12  dollars  a  cord.     What  is  the  total  cost  of  both  of  them? 

3.  A  farmer  owns  one  farm  of  286  acres,  and  another  of  1754  acres.    How 
many  acres  does  he  own  all  together  ? 

4.  A  wholesale  grocer  has  4375  cases  of  canned  corn.    To  three  customers 
he  shipped  286,  2567,  and  615  cases  respectively.     How  many  did  he  have 
left? 

5.  If  electricity  travels  on  a  wire  at  the  rate  of  288,106  miles  per  second, 
how  long  will  it  take  to  travel  144,053  miles  ? 

6.  If  one  railroad  uses  2,191,504  cross  ties  during  the  year,  and  another 
railroad  617,450  in  the  same  period  of  time,  how  many  more  ties  does  the  one 
use  than  the  other  ? 

7.  During  1918  a  citizen  bought  four  $100  Liberty  bonds,  and  two  $1000 
bonds.    What  is  the  total  of  the  sum  he  invested  in  these  bonds  ? 

Ten  graduate  students  of  the  School  of  Education  of  the  University 
of  Chicago  were  asked  to  solve  all  of  the  problems.  They  were  instructed 


NUMERALS  IN  ARITHMETICAL  PROBLEMS  3 

to  work  the  problems  rapidly  and  accurately,  and  with  pencil  and  paper 
or  without,  as  they  preferred.  They  were  urged  to  observe  faithfully 
the  arithmetic  problem-solving  attitude  from  the  beginning  of  a  problem 
to  its  solution.  After  the  answer  of  each  problem  was  recorded  the 
subjects  were  asked  to  describe  in  detail  their  experiences  while  reading 
the  problem  with  special  reference  to  the  numerals.  After  the  first 
problem  had  been  solved  and  the  reading  of  its  numerals  described, 
the  subjects  began  to  note  the  kinds  of  experiences  they  were  asked  to 
observe.  As  a  result  they  were  able  to  give  the  desired  description 
more  promptly  and  easily  with  each  successive  problem.  These  were 
recorded  and  condensed  into  tables  I-VI. 

2.       FIRST   READING   AND   RE-READING  DISTINGUISHED 

The  most  obvious  and  general  fact  noted  in  the  records  of  the  several 
subjects  was  their  clear  and  unmistakable  differentiation  of  the  reading 
of  a  problem  into  two  definite  and  distinct  phases  differing  in  time  and 
in  purpose,  namely,  the  first  reading  and  the  re-reading.  Subsequent 
sections  of  the  investigation  emphasize  the  importance  of  this  observation 
concerning  the  distinction  between  two  phases  of  the  reading  of  a  problem. 
The  general  procedure  of  each  subject  was,  first,  to  read  the  problem 
through  "to  get  the  sense"  or  "to  see  what  was  to  be  done  with  the 
numbers,"  and  secondly,  to  re-read  one  or  ah1  of  the  numerals,  and 
sometimes  also  a  few  of  the  accompanying  words.  These  re-readings  of 
the  numerals  were  for  such  purposes  as  "verification"  of  their  first 
reading,  or  the  "cultivation  of  assurance"  before  copying  the  figures  on 
paper  for  computation.  The  subject  with  one  or  two  exceptions  was 
not  aware  that  he  habitually  followed  such  a  procedure  until  he  began 
to  make  introspective  observations  of  his  habits. 

3.      PARTIAL  FIRST   READING  AND   WHOLE   FIRST   READING   OF    NUMERALS 

The  knowledge  gained  during  the  first  reading  was  found  to  be  very 
different  in  different  cases.  Subjects  sometimes  perceived  numerals  as 
merely  numerals;  sometimes  they  noted  only  the  first  digit  or  the  first 
two  digits.  At  times  they  noted  the  number  of  digits  but  did  not  attend 
to  any  one  in  particular.  Again  they  reported  a  numeral  as  large  or  as 
small,  or  as  larger  or  smaller  than  some  other  numeral.  Sometimes 
they  noted  its  location  in  the  typewritten  line.  Frequently  two  or 
more  of  these  items  were  included  in  the  general  perception.  In  all 
such  cases  as  have  been  described,  the  perception  lacks  detail  and  preci- 
sion. It  is  evidently  a  kind  of  cursory  preliminary  recognition  of  the 


HOW  NUMERALS  ARE  READ 


general  character  and  setting  of  the  numeral.  Its  value  consists  in  the 
fact  that  it  permits  the  subject  to  think  about  the  problem  without 
entering  at  first  into  the  minute  details  of  solution. 

There  were  cases,  however,  even  in  the  first  readings,  in  which  the 
subjects  gave  attention  to  the  identity  and  place  of  every  component 
digit.  In  addition  these  careful  readers  noted  also  the  character  of  the 
numeral,  observing  whether  it  was  a  whole  number  or  a  decimal.  They 
also  gained  a  notion  of  the  magnitude  of  the  numeral  as  determined  by 
the  number  of  digits.  With  each  of  the  subjects,  cases  were  found  in 
which  the  recognition  was  full  and  detailed.  Such  cases  were  recorded 
as  "whole  first  readings."  Any  reading  which  fell  short  of  complete 
detail  was  recorded  as  a  " partial  first  reading." 

The  results  of  the  introspections  are  given  in  full  in  Table  I.  Begin- 
ning at  the  top  of  the  left-hand  column,  this  table  should  be  read  verti- 
cally as  follows:  Problem  i  contains  the  two  numerals  65  and  8,  and  these 

TABLE  I 

PARTIAL  FIRST  READINGS  AND  WHOLE  FIRST  READINGS  BY  TEN  SUBJECTS  IN 
SEVEN  PROBLEMS 


Problems 

i 

2 

3 

4 

5 

6 

7 

Numerals  given  in  problems  \ 

6I 

5 
9 

386 

I7S4 

4375 
286 

288,106 
144,053 

2,191,504 
617,450 

1918 

100 
IOOO 

[ 

12 

615 

Total  number  of  readings  

20 

4O 

2O 

40 

20 

20 

IO 

20 

Partial  first  readings 

•3 

12 

IO 

31 

13 

ii 

o 

I 

Whole  first  readings  

T6 

28 

IO 

9 

7 

9 

IO 

19 

Doubtful 

I 

were  read  altogether  a  total  of  twenty  times  by  the  ten  subjects.  Of  the 
twenty  readings,  three  were  partial  first  readings,  sixteen  were  whole 
first  readings,  and  one  could  not  be  classified. 

Examination  of  Table  I  shows  that  the  numerals  in  problems  4,  5, 
and  6,  which  were  all  longer  ones  with  three  to  seven  digits,  were  read 
partially  more  than  half  of  the  times  and  accordingly  have  percentages 
of  partial  first  readings  of  50  or  more.  The  numerals  in  problems  1,2, 
and  7,  on  the  other  hand,  were  read  wholly  more  than  half  of  the  times 
and  have  percentages  of  partial  first  readings  of  only  30  or  less.  The 
longer  numerals  are  seen  to  have  been  read  partially  more  frequently, 
while  the  shorter  numerals  are  read  in  detail  more  frequently.  Attention 
should  be  called  at  this  point,  however,  to  the  fact  that  the  whole  first 
reading  of  a  numeral  does  not  necessarily  mean  that  the  numeral  will 


NUMERALS  IN  ARITHMETICAL  PROBLEMS 


not  be  re-read.  On  the  contrary  later  discussions  in  this  report  will 
show  that  almost  all  numerals  were  re-read  after  the  first  reading, 
including  even  those  which  were  read  in  detail  during  the  first  reading. 

In  order  to  bring  out  the  relation  between  partial  and  whole  readings 
and  the  character  of  the  numerals,  Table  II  was  compiled.  This  table 
shows  the  ranks  of  the  various  numerals  with  reference  to  the  frequency 
of  partial  readings.  Percentages  were  calculated  by  dividing  the  number 
of  times  a  numeral  was  partially  read  by  the  total  number  of  readings 
of  that  numeral. 

Among  the  longer  numerals  a  greater  digit-length  appears  to  cause 
a  large  percentage  of  partial  readings.  Such  a  comparison  between 
numerals  of  different  lengths  is  significant  when  the  same  number  of 
numerals  is  used  in  the  various  problems  compared.  Problems  3,  5, 

TABLE  II 
RANKS  OF  NUMERALS  ACCORDING  TO  PERCENTAGES  OF  PARTIAL  FIRST  READINGS 


Ranks 

Percentage 
of  Partial 
First 
Readings 

Description  of 
Numerals 

Numerals  Read 

I 

77 

Four  three-  to  four-digit 

437C,;   286:   2<67;  6l< 

2 

6c 

Two  six-digit 

288,106;    I44,CXa 

2 

ec 

Two  six-  to  seven-digit 

2,101X04;   617,4^0 

4" 

CO 

Two  three-  to  four-digit 

386;    1754 

c 

3O 

Four  one-  to  two-digit 

<;  o;  31  12 

6  

I? 

Two  one-  to  two-digit 

65;  8 

7  

c 

Two  familiar 

100;  1000 

8  

o 

Date 

1918 

and  6  each  have  two  numerals.  The  six  digit  numerals  of  Problem  5, 
and  the  six-  and  seven-digit  numerals  of  Problem  6  were  given  respectively 
65  per  cent  and  55  per  cent  of  partial  first  readings.  The  three-  and  four- 
digit  numerals  of  Problem  3,  on  the  other  hand,  were  given  only  50  per 
cent  of  partial  first  readings. 

Of  the  numerals  which  were  usually  given  a  whole  first  reading,  the 
date  1918  stands  out  as  different  in  character  from  other  numerals  of 
like  length.  It  is  the  only  one  which  was  never  partially  read.  The 
very  familiar  numerals  100  and  1000  with  the  dollar  sign  attached  were 
like  the  date  for  the  most  part,  in  that  they  were  read  partially  only 
once.  In  this  case,  the  partial  reading  was  revealed  by  a  mistake  made 
by  the  reader — the  numeral  100  was  read  as  i.oo  instead  of  as  100. 

The  inclusion  of  several  numerals  in  the  same  problem  appears  to 
induce  a  greater  proportion  of  partial  first  readings.  In  Problem  4, 


6  HOW  NUMERALS  ARE  READ 

where  there  are  four  numerals,  the  percentage  of  partial  readings  is  77, 
whereas  in  Problem  3,  where  only  two  numerals  appear,  the  percentage 
of  partial  readings  is  only  50,  although  the  numerals  in  both  problems 
are  of  the  same  lengths.  The  explanation  of  this  fact  seems  to  be  that 
the  subject  loses  interest  in  the  numerals  when  many  of  them  appear 
together.  Consequently  he  does  not  make  the  radical  adjustments  in 
rate  of  reading  which  would  be  necessary  for  the  careful  reading  of  a 
series  of  several  numerals. 

The  validity  of  this  explanation  is  supported  by  the  results  of  another 
comparison  of  a  similar  type  which  can  be  made  from  the  tables.  The 
numerals  in  Problems  i  and  2  are  all  one  or  two  digits  in  length.  There 
are  two  numerals  in  Problem  i,  and  four  in  Problem  2.  The  percentage 
of  partial  readings  in  Problem  i  is  15,  whereas  in  Problem  2  the  percentage 
of  partial  readings  is  30,  or  twice  as  great  as  that  in  Problem  i.  In  this 
comparison,  as  in  that  above,  where  four  numerals  of  a  certain  digit- 
length  appear  in  a  problem,  they  were  more  frequently  read  partially 
than  when  only  two  such  numerals  appear. 

The  first  numeral  in  a  problem  tends  to  be  given  a  more  careful  and 
thorough  reading  than  any  of  the  other  numerals  in  the  same  problem. 
The  basis  for  this  statement  is  found  in  the  fact  that  in  three  of  the  four 
problems  which  employ  the  longer  numerals,  the  first  numeral  receives 
a  greater  number  of  whole  first  readings  than  any  of  the  numerals  that 
follow.  According  to  the  original  tabulations,  the  first  numeral  in 
Problem  5,  288,106,  was  given  five  whole  first  readings  whereas  the 
second  numeral,  144,053,  was  given  only  two  whole  first  readings. 
A  similar  preponderance  of  whole  readings  appears  in  favor  of  the  first 
numeral  in  both  problems  3  and  4.  A  comparison  of  the  foregoing  kind 
cannot  be  drawn,  however,  between  shorter  numerals,  since  they  were 
almost  invariably  given  whole  first  readings  regardless  of  their  position 
within  the  problem. 

4.      INDIVIDUAL    SUBJECTS    CLASSIFIED    AS    PARTIAL    FIRST    READERS    AND 
AS    WHOLE   FIRST  READERS 

In  Table  III  the  ten  subjects  are  arranged  in  order  from  left  to  right 
according  to  the  number  of  their  partial  readings.  They  range  from  14 
partial  first  readings  by  G  to  no  partial  first  readings  by  Subject  H. 
The  total  in  each  case  is  19  readings.  G,  Bl,  and  S  show  a  significant 
preponderance  of  partial  readings.  H,  T,  D,  and  K,  on  the  other  hand, 
exhibit  a  preponderance  of  whole  first  readings,  each  of  the  latter  showing 
12  or  more  such  readings  out  of  a  possible  19.  These  seven  subjects 


NUMERALS  IN  ARITHMETICAL  PROBLEMS 


can  accordingly  be  classified  into  two  groups  as  partial  first  readers  and 
whole  first  readers.  The  partial  readers  read  partially  not  only  the 
three-  to  seven-digit  numerals  usually  thus  read,  but  also  several  of  the 
other  numerals  which  are  usually  read  in  detail.  Similarly  the  whole 

TABLE  III 

SUBJECTS  ARRANGED  ACCORDING  TO  NUMBER  or  PARTIAL  AND 
WHOLE  FIRST  READINGS 


SUBJ 

ECTS 

G 

Bl 

S 

P 

De 

K 

Ko 

D 

T 

H 

Partial  first  readings  
Whole  first  readings 

14 
5 

I36 

12 

7 

9 
10 

8 
ii 

7 

12 

7 

6 

3 
16 

o 

Doubtful  

I 

readers  read  in  detail  not  only  the  nine  one-  and  two-digit  numerals  and 
the  familiar  numerals  which  are  usually  so  read  but  also  several  of  the 
other  numerals  which  are  usually  read  only  partially. 

Subjects  Ko,  De,  and  P  are  not  so  distinctly  marked  off  as  the  seven 
discussed  above.  However,  since  they  show  a  preponderant  number  of 
whole  first  readings,  they  may  be  classified  as  whole  first  readers.  When 
they  are  so  classified  there  are  seven  whole  first  readers  and  only  three 
partial  first  readers.  There  were,  therefore,  more  than  twice  as  many 
whole  first  readers  as  partial  first  readers  among  the  subjects  of  this 
study. 

5.      RE-READING   OP   THE   SEVERAL  NUMERALS 

After  the  first  reading  of  a  problem  it  was  left  entirely  to  the  choice 
of  the  subject  whether  he  should  or  should  not  re-read  the  numerals  in  the 
problem.  In  all  but  a  few  cases,  which  are  classified  as  "Doubtful,"  the 
reports  of  every  subject  show  when  he  re-read  any  individual  numeral. 
Table  IV  gives,  for  each  set  of  numerals,  the  number  of  times  they  were 

TABLE  IV 
NUMBER  OF  RE-READINGS 


Problems 


i 

2 

3 

4 

5 

6 

Numerals  given  in  problems  

6I 

5 
9 
3 

386 

1754 

4375 
286 
2567 

288,106 
144,053 

2,191,504 
617,450 

1918 

IOO 
IOOO 

12 

615 

Number  of  re-readings  
Numerals  not  re-read 

ii 

g 

38 

20 

39 

i 

17 

i 

20 

0 
IO 

13 

1 

Doubtful  

2 

2 

8 


HOW  NUMERALS  ARE  READ 


re-read,  the  number  of  times  they  were  not  re-read,  and  the  number  of 
doubtful  cases.  Table  V  gives  the  ranks  of  the  several  sets  of  numerals 
according  to  the  percentages  of  re-readings.  The  percentage  of  re- 
readings  for  any  set  of  numerals  was  found  by  dividing  the  total  num- 
ber of  re-readings  which  the  numerals  of  the  set  received,  by  the  total 
number  of  re-readings  which  it  was  possible  for  them  to  have  received. 
Examination  of  Table  IV  reveals  the  fact  that  the  numerals  of  every 
set  but  one  were  very  generally  re-read.  The  longer  numerals  were 
re-read  almost  without  exception.  From  Table  V  it  is  seen  that  the 
four  sets  of  numerals  from  three  to  seven  digits  in  length  which  are  found 
in  problems  3,  4,  5,  and  6,  received  85  per  cent,  97.5  per  cent,  and  100 
per  cent,  respectively,  of  the  numbers  of  possible  re-readings.  In  two 
cases  only  were  numerals  of  this  length  reported  as  not  re-read,  and  only 

TABLE  V 
RANKS  OF  NUMERALS  ACCORDING  TO  PERCENTAGES  OF  RE-READINGS 


Ranks 

Percentage 
of 
Re-readings 

Description 
of  Numerals 

Numerals  Read 

I    r 

IOO 

Two  six-  to  seven-digit 

2.IQI  ^04.'   6l7  4.^0 

I    t 

IOO 

Two  three-  to  four-digit 

386'    17^4 

07    <? 

Four  three-  to  four-digit 

4^7^;  286;  2^67'  61=; 

QC 

Four  one-  to  two-digit 

c;    Q;    •!•    12 

5" 

8c 

Two  six-digit 

288,106:   14.4,  0^3 

6 

65 

Two  familiar 

100;  1000 

7    . 

cer 

Two  one-  to  two-digit 

6=;;  8 

8 

o 

Date 

1018 

two  cases  were  reported  as  doubtful.  Since  there  are  ten  ordinary 
numerals  of  three-  to  seven-digit  lengths,  it  was  possible  for  these 
numerals  to  have  been  re-read  a  total  of  one  hundred  times  by  the  ten 
subjects.  Of  this  number  of  possible  re-readings  the  three-  to  seven-digit 
numerals  actually  were  re-read  ninety-six  times. 

Even  the  short  one-  and  two-digit  numerals  and  the  familiar  numeri- 
cals  were  very  generally  re-read,  although  usually  they  had  been  given 
whole  first  readings.  Each  of  these  sets  of  numerals,  with  the  excep- 
tion of  the  familiar  date  numeral,  was  re-read  50  per  cent  or  more 
of  the  possible  times.  The  one-  and  two-digit  numerals,  in  problems 
which  contain  only  two  numerals,  were  re-read  55  per  cent  of  the 
possible  times;  but  the  numerals  of  this  same  length,  in  problems 
which  contain  four  numerals,  were  re-read  95  per  cent  of  the  possible 
times.  The  familiar  numerals  with  the  dollar  sign  attached  were  re-read 
65  per  cent  of  the  possible  times.  The  only  numeral  never  re-read  is  the 


NUMERALS  IN  ARITHMETICAL  PROBLEMS 


familiar  date  1918,  which  was  not  necessary  in  any  way  to  the  solution 
of  the  problem  in  which  it  appears. 

6.       RE-READING   BY   INDIVIDUAL   SUBJECTS 

The  very  large  percentages  of  re-readings  of  the  several  sets  of 
numerals  imply  that  most  of  the  individual  subjects  are  persistent 
re-readers.  Examination  of  Table  VI,  in  which  the  facts  for  each  indi- 
vidual subject  are  displayed,  proves  that  such  is  the  case.  Subjects 
G,  Bl,  and  S  who  had  read  most  of  the  numerals  only  partially  during 
the  first  reading,  proceeded  to  re-read  the  numerals  before  they  began  to 
solve  the  problems.  They  re-read  not  only  the  numerals  at  which  they 
had  merely  glanced  the  first  time,  but  also  those  which  they  had  read  in 
detail  at  first  reading.  None  of  these  three  subjects  showed  a  number 
of  numerals  not  re-read  equal  to  the  number  of  numerals  which  he  had 
read  in  detail  at  first  reading. 

TABLE  VI 

NUMBER  OF  RE-READINGS  BY  VARIOUS  SUBJECTS 


SUB. 

[ECTS 

G 

Bl 

S 

P 

De 

K 

Ko 

D 

T 

H 

Number  of  re-readings  
Numerals  not  re-read 

16 
3 

18 

i 

16 
3 

'1 

17 

i 

.17 
2 

18 

16 
3 

12 

5 

18 

Doubtful 

I 

2 

The  seven  subjects  who  were  classified  as  whole  first  readers  re-read 
practically  as  many  of  the  numerals  as  the  partial  first  readers.  Subject 
H,  who  gave  all  of  the  numerals  a  whole  first  reading,  re-read  as  many 
numerals  as  any  partial  first  reader.  The  smallest  numbers  of  numerals 
re-read  are  found  in  the  records  of  the  whole  first  readers,  T  and  P. 
These  same  subjects  have  the  largest  numbers  of  numerals  not  re-read. 
Besides  the  familiar  numerals  and  the  one-  and  two-digit  numerals,  T 
did  not  re-read  the  numeral  4375  and  P  did  not  re-read  288,106. 

At  this  time  attention  should  be  called  to  the  fact,  which  will  be 
elaborated  in  chapter  viii,  that  the  re-reading  of  numerals  appears  to 
be  very  closely  connected  with  copying  them  on  paper  for  computation. 
With  rare  exceptions  the  subjects,  when  solving  the  problems  of  this 
study,  followed  the  procedure  of  copying  the  numerals  and  computing 
with  pencil.  In  the  chapter  referred  to  above,  however,  it  was  found 
that  several  subjects,  including  G  and  H  of  the  present  study,  solved  the 
problems  which  were  read  in  the  second  part  of  the  investigation  "men- 


10  HOW  NUMERALS  ARE  READ 

tally"  and  directly  from  the  text.  Such  subjects  when  solving  problems 
"  men  tally"  did  not  re-read  the  numerals. 

After  each  subject  had  solved  all  of  the  problems,  he  was  asked  to 
describe  any  individual  attitudes  or  previous  experiences  which  he 
believed  had  affected  in  an  important  way  his  methods  of  reading 
numerals  in  arithmetical  problems.  Three  of  the  subjects  gave  descrip- 
tions which  threw  light  upon  their  procedures  as  reported  above. 
Subject  T  stated  that  he  has  had  from  his  early  school  days  unusual  ability 
in  retaining  by  visual  memory  both  long  and  short  numerals  which  he  had 
read.  This  ability  had  recently  undergone  intensive  training  in  the 
form  of  much  reading  and  copying  of  army  serial  numbers,  which  his 
work  as  company  clerk  in  the  army  required.  He  believes  his  habit  is 
to  read  numerals  in  detail  wherever  he  sees  them,  and  that  he  could  have 
solved  perhaps  all  of  the  seven  problems  immediately  after  the  first 
reading  without  looking  at  the  numerals  again.  He  goes  on  to  say, 
however,  that  notwithstanding  his  careful  first  reading  of  the  numerals, 
he  is  in  the  habit  of  re-reading  them  before  beginning  computation 
"in  order  to  be  sure." 

Subject  H,  who  gave  every  numeral  a  whole  first  reading,  states 
that  early  difficulties  with  arithmetical  problems  caused  him  to  develop 
habits  of  great  caution  in  reading  and  solving  them.  He  describes  his 
procedure  as  beginning  with  a  careful,  complete  reading  of  every  numeral 
in  a  problem  when  he  first  comes  to  it.  He  then  returns  to  re-read 
every  numeral  and  sometimes  some  of  the  words. 

Subject  G,  whose  record  presents  the  largest  number  of  partial  first 
readings  of  the  numerals,  reports  that  he  intends  to  obtain  "only  a 
general  idea"  of  the  numerals  from  the  first  reading,  especially  if  they 
are  long  ones.  He  explains  that  he  chose  this  attitude  toward  the 
numerals  after  he  had  learned  through  experience  that  he  was  unable 
to  recall  them  for  computation  and  "had  to  go  back  for  them  anyhow." 

7.     SUMMARY 

A  summary  of  the  results  of  the  first  preliminary  study  on  how 
numerals  in  arithmetical  problems  are  read  includes  the  following 
points: 

i.  The  subjects  distinguished  two  phases  in  the  reading  of  problems, 
namely,  a  first  reading  and  a  re-reading.  The  purpose  of  the  first 
reading  is  to  discover  the  conditions  of  the  problem,  while  that  of  the 
re-reading  is  to  perceive  the  numerals  accurately  for  use  in  computation. 


NUMERALS  IN  ARITHMETICAL  PROBLEMS  II 

2.  Two  ranges  of  perception  of  numerals  during  the  first  reading 
are  distinguished,  namely,  whole  first  reading  and  partial  first  reading. 

3.  Shorter  numerals  and  very  familiar  numerals  more  frequently 
receive  whole  first  readings,  whereas  longer  numerals  more  frequently 
receive  partial  first  readings. 

4.  The  first  numerals  in  problems  which  have  numerals  of  three  to 
seven  digits  in  length,  are  commonly  given  whole  first  readings. 

5.  When  as  many  as  four  numerals  appear  in  a  problem  they  receive 
a  greater  proportion  of  partial  first  readings  than  in  those  cases  in  which 
only  two  numerals  of  the  same  digit-length  appear. 

6.  Subjects  differ  widely  in  their  habits.     Some  are  predominantly 
whole  first  readers,  others  are  partial  first  readers. 

7.  Numerals  of  all  lengths  and  types,  when  they  are  used  in  computa- 
tion, are  very  generally  re-read  for  computation. 

8.  All  subjects  persistently  re-read  numerals  when  they  begin  to  use 
them  in  computation. 


CHAPTER  III 

RANGE  OF  CORRECT  RECALL  OF  NUMERALS  AFTER  THE  FIRST 
READING— SECOND  PRELIMINARY  STUDY 

I.      DESCRIPTION   OF   THE   STUDY 

The  purpose  of  the  second  preliminary  study  was  to  obtain  further 
information  as  to  the  nature  of  the  readings  of  the  numerals  during  the 
first  reading  of  a  problem.  The  general  plan  followed  for  the  accomplish- 
ment of  this  purpose  was  to  have  the  subjects  report  all  of  the  details 
of  the  numerals,  which  they  were  able  to  recall  immediately  after  the 
first  reading. 

The  subjects  were  seven  graduate  students  in  the  School  of  Education 
of  the  University  of  Chicago.  Of  the  seven  subjects,  only  one,  Subject 
G,  had  served  in  the  first  preliminary  study.  The  materials  which  they 
read  were  twelve  simple  arithmetical  problems  similar  to  those  used  in 
the  first  preliminary  study.  Six  of  the  problems  were  work  problems 
in  the  sense  that  each  of  them  was  worked  until  an  answer  was  found. 
No  questions,  however,  were  asked  concerning  the  numerals  of  these 
problems.  Their  function  was  to  help  the  subjects  maintain  throughout 
the  study  the  problem-solving  attitude  by  actually  solving  problems. 
The  readings  of  the  other  six  problems  were  the  bases  of  the  data  which 
were  used  in  the  study.  In  the  context  of  these  problems  a  total  of 
sixteen  numerals  varying  in  length  from  one  to  seven  digits  was  included. 
Two  or  three  numerals  were  placed  hi  each  of  five  of  the  problems. 
In  Problem  D,  however,  four  numerals  appeared  together.  Numerals 
of  similar  length  were  placed  in  the  same  problem,  as  was  done  in  the 
problems  of  the  first  preliminary  study.  All  of  the  problems  were 
presented  in  typewriting. 

The  problems  which  were  read  for  data  are  as  follows: 

PROBLEM  A 

At  43  cents  a  dozen,  what  will  2  dozen  eggs  cost  ? 

PROBLEM  B 

If  a  man  buys  5  tons  of  coal  for  45  dollars,  what  will  he  have  to  pay  for 
8  tons  ? 

PROBLEM  C 

A  farmer  owns  one  farm  of  246  acres,  and  another  of  1754  acres.  How 
many  acres  does  he  own  altogether  ? 

12 


CORRECT  RECALL  OF  NUMERALS  AFTER  FIRST  READING       13 

PROBLEM  D 

A  wholesale  merchant  had  on  hand  1000  cases  of  canned  corn.  From 
three  factories  he  bought  1276,  91  and  718  cases  respectively.  How  many  did 
he  then  have  ? 

PROBLEM  E 

If  one  railroad  uses  2,981,534  cross  ties  during  the  year,  and  another  rail- 
road 617,453  in  the  same  period  of  time,  how  many  more  ties  does  the  one  use 
than  the  other  ? 

PROBLEM  F 

If  one  man  can  do  a  piece  of  work  in  10  days,  and  another  can  do  the  same 
work  in  9  days,  what  should  be  the  wages  of  the  second,  if  the  wages  of  the 
first  are  3  dollars  ? 

The  subjects  were  informed  that  they  would  be  expected  to  solve 
most  of  the  problems,  but  that  they  would  be  interrupted  in  their  proce- 
dure in  some  of  them  by  the  placing  of  a  cardboard  screen  over  the  text 
which  was  being  read.  They  were  told  that  they  would  not  be  informed 
beforehand  as  to  which  problems  were  to  be  solved  and  which  were  to 
be  covered  with  the  screen.  Accordingly  they  were  urged  to  maintain 
their  normal  problem- solving  attitude  toward  every  problem  that  was 
presented. 

The  time  chosen  for  the  placing  of  the  screen,  when  it  was  to  be 
used,  was  the  moment  when  the  subject  completed  the  first  reading  of 
the  problem.  The  subjects  were  therefore  instructed  to  signal  by  a 
slight  movement  of  the  left  hand  the  moment  at  which  they  were 
completing  the  first  reading.  This  they  were  able  to  do  without  embar- 
rassment after  a  brief  training  period  with  practice  problems.  The 
screen  was  used  only  with  those  six  problems  that  carried  in  their 
context  the  numerals  which  were  to  be  studied. 

Immediately  after  the  screen  was  placed  over  one  of  these  problems 
the  subject  was  asked  to  give  in  detail  all  of  the  items  of  the  numerals 
which  he  could  recall.  After  having  proceeded  in  this  manner  with  the 
first  problem  which  was  to  be  studied  in  this  way,  the  subject  understood 
what  kinds  of  details  concerning  the  numerals  were  desired,  and  with 
subsequent  problems  easily  and  promptly  gave  such  of  the  details  as  he 
could  recall. 

The  results  are  reported  in  tables  VII  and  VIII  under  five  classifica- 
tions. The  classification,  complete,  signifies  that  every  digit  of  the 
numeral  was  recalled  in  its  true  identity  and  place.  The  second  classifica- 
tion includes  correct  recall  of  the  digit-length  of  the  numeral  and  the 
identity  and  place  of  at  least  its  first  two  digits.  The  third  classification 


HOW  NUMERALS  ARE  READ 


includes  correct  recall  of  the  digit-length  and  the  identity  of  the  first 
digit;  and  the  fourth  classification  includes  correct  recall  of  the  digit- 
length  of  the  numeral.  The  fifth  classification  includes  those  cases  in 
which  the  subjects  were  able  to  recall  nothing  more  of  a  numeral  than  its 
presence  in  the  problem. 

2.      RANGE   OF  RECALL  OF  THE   SEVERAL  NUMERALS 

The  range  of  correct  recall  from  the  first  readings  of  each  individual 
numeral  is  displayed  in  Table  VII.  The  point  which  stands  out  most 
strikingly  after  a  survey  of  the  table  is  that  almost  invariably  some  item 
from  the  first  reading  of  every  numeral  is  correctly  recalled.  A  glance 

TABLE  VII 
RANGE  OF  CORRECT  RECALL  OF  NUMERALS  FROM  FIRST  READING  OF  PROBLEMS 


ONE-  AND  TWO-DIGIT 

NUMERALS 

THREE-  TO  SEVEN-DIGIT  NUMERALS 

i 

!< 

Prob- 
lem 

Prob- 
lem 

1* 

Prob- 
lem 

Problem 

Problem 

gW 

so 

hj 

I 

B 

F 

*< 

C 

D 

E 

i_3 

< 

Numerals   read  in 

problems  
Total    number    of 

43 

2 

5 

45 

8 

10 

9 

3 

246 

I7S4 

1000 

1276 

9i 

7i8 

2,981,534 

6i7,453 

.... 

readings      given 

each  numeral  by 

all  subjects  

7 

7 

7 

7 

7 

6 

6 

6 

S3 

6 

6 

6 

6 

6 

6 

7 

7 

50 

103 

Range    of    correct 

recall  of  numer- 

als: Complete  .  . 

5 

6 

6 

6 

7 

6 

6 

6 

48 

6 

i 

5 

i 

0 

i 

0 

o 

14 

62 

First  two  digits 

and  digit-length 

5 

6 

6 

17 

6 

4 

5 

3 

0 

i 

i 

i 

21 

38 

First    digit    and 

digit-length..  .  . 

6 

6 

6 

7 

7 

6 

6 

6 

50 

6 

5 

5 

4 

2 

i 

6 

3 

32 

82 

Digit-length  
Merely     noticed 

6 

i 

6 

0 

6 

0 

7 

0 

7 

0 

6 

0 

6 

0 

6 

0 

50 

I 

6 
O 

6 

o 

5 

i 

6 
o 

4 

0 

3 

2 

7 

0 

7 

0 

44 
3 

94 
4 

at  the  "totals"  column  at  the  extreme  right  shows  that  some  item  of  the 
numerals  in  94  of  the  103  total  number  of  readings  was  recalled.  In 
five  cases  only,  the  numeral  was  no.t  even  "merely  noticed."  Such 
frequency  of  recall  implies  that  in  the  minds  of  subjects  confronted  with 
arithmetical  problems  to  be  solved  the  numerals  hold  a  place  of  unique 
significance  among  the  other  elements  of  the  problems,  and  in  con- 
sequence are  noticed  almost  invariably. 

The  range  of  recall  which  most  frequently  follows  this  habitual 
notice  of  the  numerals  also  stands  out  clearly  from  this  table.  This 
item  is  the  correct  digit-length  of  the  numerals.  It  was  recalled  from  a 
very  great  majority  of  the  readings,  namely,  from  94  of  the  total  103 
cases.  All  of  the  exceptions  occur  in  Problem  4  where  four  numerals  of 


CORRECT  RECALL  OF  NUMERALS  AFTER  FIRST  READING        15 

three  different  digit-lengths  occur.  In  such  a  situation  it  was  an  easy 
matter  for  the  length  of  one  numeral  to  be  confused  with  that  of  another. 
Evidently  the  number  of  digits  in  a  numeral  is  a  point  of  exceptional 
interest  to  the  readers.  This  is  not  difficult  to  understand  in  view  of  the 
fact  that  the  number  of  digits  in  a  numeral  is  certainly  one  of  the  most 
highly  significant  indications  of  its  value. 

With  the  shorter  numerals,  complete  recall  was  achieved  in  almost 
every  instance.  In  the  left-hand  section  of  Table  VII  it  is  shown  that 
such  is  the  case  in  48  instances  of  a  total  of  53.  Apparently  no  greater 
demands  were  made  upon  the  attention  of  the  subjects  by  the  reading 
of  one-  and  two-digit  numerals  than  practically  all  of  them  were  both 
able  and  willing  to  meet.  It  is  also  apparent  that  the  effort  which  was 
habitually  used  in  reading  the  short  numerals  so  firmly  fixed  them  in 
memory  that  they  were  able  to  be  recalled  completely. 

The  longer  numerals  on  the  contrary  do  not  exhibit  such  large 
preponderances  in  the  higher  ranges  of  recall  as  are  exhibited  by  the 
shorter  numerals.  In  comparatively  few  instances  the  longer  numerals 
were  completely  recalled.  The  explanation  is  believed  to  be  due  to  two 
facts:  first,  that  longer  numerals  according  to  common  experience  are 
more  difficult  to  recall  than  shorter  numerals;  and  second,  that  in  most 
cases  the  longer  numerals  were  only  partially  read  during  the  first 
reading  of  the  problems. 

When  reading  the  longer  numerals  the  subjects  evidently  gave  more 
emphatic  attention  to  the  first  one  or  two  digits  than  to  any  other 
digits.  In  more  than  half  of  the  cases  the  first  digit  was  recalled,  and 
in  slightly  less  than  half  of  the  cases  the  first  two  digits  were  recalled. 
In  only  three  instances  were  three  or  more  digits  retained,  when  recalls 
of  the  familiar  numeral  1000  and  of  the  numeral  246,  which  is  found  in 
the  peculiar  place  of  "first"  numeral  in  Problem  C,  are  excepted.  The 
chief  explanation  of  this  greater  emphasis  on  the  first  digits  probably 
lies  in  the  general  habit  of  attacking  printed  matter  from  the  left.  It  is 
also  possible  that  the  adult  subjects  of  this  study  had  learned  empirically 
the  greater  value  of  the  first  digit  of  a  numeral  and  its  greater  significance 
to  the  solving  of  the  problem,  and  therefore  had  formed  the  habit  of 
paying  special  attention  to  it  when  reading  problems. 

The  first  numeral  of  a  problem  seems  to  receive  more  careful  attention 
during  the  first  reading  than  any  of  the  other  numerals  in  the  problem. 
More  details  of  the  first  numeral  are  correctly  recalled  than  of  other 
numerals.  Such  a  comparison  can  be  made  between  the  longer  numerals 
only,  because,  as  has  been  pointed  out  in  previous  paragraphs  of  this 


i6 


HOW  NUMERALS  ARE  READ 


study,  the  shorter  numerals  almost  invariably — and  therefore  quite 
without  regard  to  position  in  the  problem — are  read  closely  enough  for 
complete  recall.  The  longer  numerals  appear  in  problems  C,  D,  and  E. 
In  each  of  these  problems  more  details  of  the  first  numeral  are  recalled. 
Problem  C  will  serve  as  an  illustration.  Here  the  first  numeral,  246,  is 
completely  recalled  in  all  cases,  while  the  second  numeral  is  so  recalled 
only  once.  In  Problem  D  the  numeral  1276  is  considered  the  first 
numeral  rather  than  1000  because  of  the  peculiar  character  of  the  latter 
numeral.  The  higher  ranges  of  recall  of  first  numerals  are  probably 
due  in  part  to  the  advantages  of  "initial"  position.  By  virtue  of  this 
position  they  would  tend  to  be  more  vividly  impressed  upon  the  memories 
of  the  readers.  In  additon  to  this  advantage  it  appears  probable  that 
the  adult  subjects  of  this  study  have  learned  empirically  to  pay  greater 
attention  to  the  first  numeral.  By  so  doing  they  would  be  able  to  make 
special  use  of  it,  not  only  in  determining  the  conditions  of  the  problem, 
but  also  as  a  base  in  reaching  decisions  concerning  the  relations  of  the 
numerals  to  each  other. 

The  peculiar  quality  of  the  familiar  numeral  1000  again  distinguishes 
it  from  other  numerals  of  the  same  length.  In  every  case  but  one  it  was 
recalled  completely.  This  single  exception  was  due  to  an  unusual  case 
of  confusion  on  the  part  of  the  subject,  which  caused  him  to  forget  even 
the  sense  of  che  problem. 

3.      RANGE   OF  RECALL — BY  THE   SEVERAL   SUBJECTS 

Certain  subjects  recall  much  more  of  the  numerals  than  others. 
The  significant  differences  between  them  occur  in  the  higher  ranges  of 
recall  and  with  the  longer  numerals.  These  differences  are  displayed  in 
detail  in  Table  VIII.  The  several  individuals  divide  themselves  into 
two  general  groups  according  as  they  reported  relatively  many  or 

TABLE  VIII 

VARYING  RANGES  OF  CORRECT  RECALL  OF  THREE-  TO  SEVEN-DIGIT  NUMERALS  BY 

THE  SEVERAL  SUBJECTS 


SUBJECT. 

> 

Hb 

R 

Bak 

Th 

L 

G 

C 

Number  of  numerals  read  by  subjects  
Range  of  correct  recall  of  numerals: 
Complete                                            

8 
4 

8 

2 

8 

2 

4 

2 

8 

2 

6 

i 

8 

i 

First  two  digits  and  digit-length  

7 

4 

4 

2 

2 

i 

i 

First  digit  and  digit-length         

7 

7 

6 

3 

3 

2 

4 

Digit-length 

8 

7 

8 

4 

5 

5 

7 

Merely  noticed              

o 

i 

o 

0 

I 

o 

i 

CORRECT  RECALL  OF  NUMERALS  AFTER  FIRST  READING        17 

relatively  few  of  the  longer  numerals  in  the  higher  ranges  of  recall. 
Subjects  Hb,  R,  Bak,  and  Th  are  included  in  the  first  group,  and  L,  G, 
and  C  constitute  the  second  group.  The  contrast  between  Hb  of  the 
first  group  and  G  of  the  second  is  striking.  The  former  completely 
recalls  half  of  the  longer  numerals  and  the  first  two  digits  of  seven  of  the 
eight  longer  numerals  read,  while  the  latter  recalls  completely  only  one 
longer  numeral  and  the  first  two  digits  of  only  one. 

Differences  between  the  two  individuals  in  first-reading  attitudes 
seem  to  account  for  the  large  differences  exhibited  by  them  in  the 
ranges  of  recall.  Subject  Hb  is  a  pronounced  whole  first  reader.  He 
intends  to  " grasp"  all  of  a  numeral  when  he  first  reads  it.  Subject  G, 
on  the  other  hand,  is  a  striking  example  of  the  type  of  partial  first 
readers.  His  purpose  during  the  first  reading,  in  so  far  as  the  numerals 
are  concerned,  is  to  obtain  only  a  " general  idea." 

4.      FURTHER  EVIDENCE  AS  TO  THE  PURPOSE   OF  FIRST   READING 

Further  evidence  is  found  in  this  study  in  support  of  the  conclusion 
presented  in  the  first  preliminary  study  that  the  main  purpose  of  the 
first  reading  is  to  find  the  conditions  of  the  problem  in  order  to  know 
how  to  proceed  with  solving.  In  nearly  all  instances  the  subjects  in 
this  study  were  able  to  indicate  a  correct  procedure  for  the  solution  of 
any  problem  after  the  first  reading.  They  were  able  to  do  this  even  in 
the  many  instances  where  they  could  recall  nothing  more  of  the  numerals 
than  their  digit-lengths. 

5.      ITEMS   OF  RECALL  NOT  INCLUDED  IN  THE  CLASSIFICATIONS 

The  classification  scheme  used  in  this  study  does  not  include  every 
item  concerning  the  numerals  which  was  reported.  In  many  cases 
subjects  reported  correctly  the  line  of  the  problem  in  which  a  numeral 
appeared.  In  several  cases  they  recalled  its  approximate  location 
within  the  line.  No  items  incorrectly  recalled  are  included  in  the 
classifications.  Several  such  items  were  reported.  In  general  they 
followed  the  types  of  errors  which  would  be  found  in  any  study  of  errors 
in  the  reading  of  numerals  in  arithmetical  problems. 

6.      SUMMARY  OF  CONCLUSIONS 

The  results  of  the  second  preliminary  study  on  the  range  of  correct 
recall  of  numerals  after  first  reading  may  be  summarized  as  follows: 
(i)  Some  item  of  almost  every  numeral  is  recalled.  (2)  The  digit  length 
of  numerals  is  recalled  almost  invariably.  (3)  The  shorter  numerals 
and  the  familiar  numeral  1000  are  completely  recalled  almost  invariably. 


1 8  HOW  NUMERALS  ARE  READ 

(4)  The  first  one  or  two  digits  of  longer  numerals  are  recalled  in  a  majority 
of  cases.  (5)  The  first  numeral,  in  problems  which  include  numerals  of 
the  greater  lengths,  is  more  frequently  recalled  than  any  other  numeral 
in  the  problem.  (6)  The  subjects  divided  themselves  into  two  groups 
according  as  they  recalled  in  the  higher  ranges  large  or  small  proportions 
of  the  longer  numerals.  (7)  Further  evidence  appears  in  support  of  the 
previous  conclusion  that  the  main  purpose  of  the  first  reading  of  a 
problem  is  to  learn  its  conditions. 


CHAPTER  IV 

ANALYSIS  OF  THE  RE-READING  OF  NUMERALS  IN  ARITHMETICAL 
PROBLEMS— THIRD  PRELIMINARY  STUDY 

I.      DESCRIPTION   OF   THE    STUDY 

The  distinction  was  drawn  between  the  first-reading  and  the  re- 
reading phases  of  the  reading  of  arithmetical  problems  in  the  first 
preliminary  study.  The  general  purpose  of  re-reading  as  stated  was 
"to  perceive  the  numerals  accurately  for  computation."  The  present 
study  was  designed  to  give  further  description  of  the  purposes  of  the 
subjects  and  of  their  activities  with  the  numerals  during  the  re-readings. 
The  general  method  which  was  used  to  obtain  the  data  was  that  of 
introspective  observation  on  the  part  of  adult  readers. 

The  readers  were  four  graduate  students  in  the  School  of  Education 
of  the  University  of  Chicago.  One  of  them,  Subject  S,  had  read  the 
problems  of  the  first  preliminary  study.  None  of  the  others  served  as 
subjects  in  any  other  study  of  the  investigation.  They  were  asked  to 
solve  the  five  simple  arithmetical  problems  which  were  later  used  as 
reading  materials  in  the  eye-movement  studies  and  which  are  described 
in  detail  in  chapter  vi.  Each  subject  was  given  pencil  and  paper  and 
told  that  he  might  use  them  in  solving  the  problems  or  not  use  them,  as 
he  chose.  Before  the  beginning  of  the  experiment  the  subjects  were 
informed  concerning  the  first-reading  and  re-reading  phases  of  the 
reading  of  problems.  At  the  conclusion  of  the  experiment  each  of  the 
subjects  was  of  the  opinion  that  his  reading  of  arithmetical  problems 
habitually  followed  these  phases,  and  that  the  information  given  con- 
cerning them  had  not  caused  him  to  vary  from  his  normal  procedure. 

The  subjects  were  instructed  to  attack  each  problem  immediately 
when  it  was  presented  and  to  proceed  with  it  in  accordance  with  their 
normal  problem-solving  attitude.  They  were  to  press  a  conveniently 
placed  telegraph  key  at  the  instant  of  beginning  to  read  and  continue 
the  pressure  throughout  the  first  reading.  Immediately  at  the  con- 
clusion of  the  first  reading  the  key  was  released.  Thereafter  whenever 
the  attention  of  the  subject  was  directed  to  the  re-reading  of  any  item 
from  the  text  of  the  problem,  the  key  was  pressed  and  held,  until  atten- 
tion was  directed  away  from  the  text  whereupon  the  key  was  immediately 
released.  The  effect  of  this  practice  was  to  secure  a  separate  record  for 
each  of  the  one  or  more  acts  of  re-reading  from  the  text  of  a  problem. 

19 


20  HOW  NUMERALS  ARE  READ 

Every  pressure  and  release  of  the  key  was  recorded  on  a  smoked- 
paper  record  sheet  which  was  moving  on  two  kymograph  drums.  The 
duration  of  each  pressure  on  the  key  was  measured  in  seconds  by  the  use 
of  a  chronometer  which  was  so  placed  that  its  marker  recorded  the  time 
intervals  on  the  record  sheet  side  by  side  with  the  records  from  the  key. 
A  brief  period  of  training  with  practice  problems  in  this  procedure  was 
necessary  in  order  to  enable  the  subjects  to  follow  the  procedure  correctly 
and  easily.  Immediately  after  the  solving  of  a  problem,  and  with  its 
text  before  them  for  reference,  the  subjects  were  asked  to  report  the 
words  or  numerals  in  the  text  of  the  problem,  upon  which  their  attention 
was  directed  at  each  separate  re-reading.  This  they  were  able  to  do  with 
promptness  and  certainty.  The  reports  of  the  subjects  and  the  time 
records  from  the  kymograph  are  presented  in  tables  IX  and  X. 

The  reading  of  Problem  2  by  Subject  Ba  will  serve  as  an  illustration 
of  the  experimental  procedure.  Ba  began  to  read  the  problem  imme- 
diately when  it  was  placed  before  him  and  at  the  same  moment  he  pressed 
the  key.  The  instant  he  finished  the  first  reading  of  the  problem, 
which  required  a  time  interval  of  7.6  seconds,  he  released  the  key. 
Without  delay  he  turned  his  attention  to  the  numeral  357  in  the  text  of 
the  problem  and  immediately  pressed  the  key.  During  an  interval  of 
1.4  seconds  he  re-read  this  numeral.  He  then  directed  his  attention  to 
the  sheet  of  paper  on  which  he  intended  to  copy  the  numeral  and  at  the 
same  time  released  the  key.  When  357  was  copied  he  turned  his  atten- 
tion to  the  numeral  1643,  pressing  the  key  at  the  same  instant.  During 
an  interval  of  2.4  seconds  he  re-read  this  numeral.  When  the  re-reading 
of  1643  was  completed  he  looked  to  the  copy  sheet  to  copy  the  numeral 
and  at  that  moment  released  the  key.  This  done,  once  more  he  glanced 
at  the  problem,  simultaneously  pressing  the  key,  and  fixed  his  attention 
upon  the  last  sentence  for  .2  of  a  second.  At  the  conclusion  of  this 
interval  he  released  the  key  and  was  ready  to  proceed  with  solving  the 
problem. 

The  numerals  or  words  read  at  each  re-reading  from  the  problems 
are  given  for  every  subject  in  Table  IX.  The  time  in  seconds  required 
for  re-reading  the  numerals  or  words  is  given  under  the  numerals  or  words 
in  every  case.  Beginning  at  the  top  of  the  table  the  first  left-hand 
column  reads  that  two  re-readings  were  given  to  items  from  Problem  i 
which  contains  the  numerals  47  and  2.  The  item  from  the  text  of  the 
problem  which  was  read  by  the  first  re-reading  was  the  numeral  47,  and 
the  duration  of  this  re-reading  was  2.4  seconds.  At  the  second  re-reading 
the  numeral  2  was  read  and  the  time  required  for  this  second  re-reading 


RE-READING  OF  NUMERALS  IN  ARITHMETICAL  PROBLEMS      21 


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Numerals  read  in  problems  I 

Ordinal  number  of  re-readings  

I     13         '"o         '•  TJ           '•      *a 

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22  HOW  NUMERALS  ARE  READ 

was  again  2.4  seconds.  The  sum  of  the  durations  of  the  two  re-readings 
from  the  problem,  i.e.,  the  total  re-reading  time  for  the  problem  was, 
therefore,  2.4  seconds+2.4  seconds  or  4.8  seconds.  This  last  detail  of 
information  appears  in  Table  X,  in  the  first  left-hand  column,  and  in 
the  upper  row. 

2.      OBJECTS   AND   NATURE    OF   THE    RE-READINGS 

The  point  which  stands  out  most  clearly  in  Table  IX  is  that  numerals 
were  the  objects  of  the  re-readings  almost  invariably.  Only  three 
instances  appear  in  the  entire  number  of  re-readings  in  which  the  objects 
of  the  re-readings  were  words.  It  is  also  clearly  apparent  that  the 
numerals  of  the  problems  were  almost  invariably  re-read.  In  five 
instances  only,  numerals  were  not  re-read. 

These  facts  serve  as  additional  evidence  in  support  of  the  conclusion, 
which  was  drawn  in  a  previous  section,  that  the  numerals  are  of  a  nature 
which  clearly  distinguishes  them  from  the  other  contextual  elements  of 
arithmetical  problems  and  which  causes  them  to  make  unusual  demands 
upon  the  attention  of  readers. 

Further  light  is  thrown  upon  the  procedures  of  subjects  with  the 
numerals  by  a  review  of  the  original  reports.  These  reports  show  that 
every  numeral  which  was  re-read  was  copied  on  the  computation  paper 
immediately  after  it  was  re-read.  In  all  of  these  cases  paper  and  pencil 
were  used  in  solving  the  problems.  In  such  cases  copying  the  numerals 
is  one  of  the  earlier  moves  in  the  total  process  of  solving  the  problem. 
Re-reading  the  numerals  appears  in  most  cases  to  have  been  a  necessary 
step  preliminary  to  copying  them.  It  is,  therefore,  proper  to  speak  of 
such  re-reading  as  re-reading  for  copying. 

The  number  of  readings  which  was  required  for  the  re-reading  of 
a  numeral  for  copying  was  usually  one.  Numerals  varying  in  length 
from  one  to  seven  digits  were  thus  re-read  at  one  reading.  To  none  of 
the  numerals  of  one-  to  four-digit  lengths  was  more  than  one  re-reading 
given.  To  several  of  the  six-  and  seven-digit  numerals,  on  the  other 
hand,  two  re-readings  each  were  given.  In  these  cases  the  first  three  or 
four  digits  of  the  numeral  were  re-read  and  copied  as  a  group,  after  which 
the  remaining  three  digits  were  re-read  and  copied  as  a  second  group. 
In  two  instances,  the  two  numerals  of  Problem  2  were  copied  from  one 
re-reading. 

3.      DURATION   OF   RE-READINGS 

The  duration  of  the  first  reading  and  the  sum  of  the  durations  of 
the  re-readings  are  given  for  the  reading  of  each  of  the  several  problems 


RE-READING  OF  NUMERALS  IN  ARITHMETICAL  PROBLEMS      23 


by  each  of  the  individual  subjects  in  Table  X.  Examination  of  the 
data  shows  that  numerals  of  greater  lengths  required  greater  total 
re-reading  times  than  numerals  of  lesser  lengths.  The  total  re-reading 
time  in  the  data  for  each  individual  subject  increases  gradually,  in  most 
cases,  from  the  relatively  small  total  time  required  to  re-read  the  short 
numerals  of  Problem  i  to  the  relatively  greater  time  required  to  re-read 
the  long  numerals  of  problems  3  and  5.  The  numerals  of  Problem  4, 
which  offers  four  numerals  for  re-reading,  received  in  most  cases  a  greater 
total  re-reading  time  than  the  numerals  of  any  other  of  the  five  problems, 
none  of  which  offers  more  than  two  numerals  for  re-reading. 

TABLE  X 

NUMBER  OF  SECONDS  REQUIRED  FOR  FIRST  READING  AND  FOR 
RE-READING  OF  PROBLEMS 


Problems  

i 

2 

3 

4 

5 

Numerals  read  | 

47 

2 

357 
1643 

243,987 
21,765 

1000;  1276 
91;  817 

1,918,564 
6i7,453 

Subject  Ba  — 
First  reading  time                

2  .4 

7-6 

7  -4 

10.4 

9-4 

4  8 

4  ° 

8  o 

18.6 

12.4 

Subject  Gl  — 
First  reading  time                

2    O 

4.6 

6.6 

5  8 

6.8 

Total  re-reading  time  

0.8 

2.0 

3-4 

6    2 

5.0 

Subject  S— 
First  reading  time 

i  8 

4.  2 

7  -° 

12    8 

6.6 

Total  re-reading  time  

1.4 

3-6 

3  o 

4.0 

Subject  Wm  — 

3  8 

5  6 

10.6 

20  o 

10.8 

Total  re-reading  time  

4-  2 

7-4 

10    0 

6.8 

The  total  re-reading  time  of  a  problem  is  in  most  cases  shorter  than 
the  first  reading  time  of  the  problem.  The  obvious  explanation  lies  in 
the  comparatively  small  amount  of  work  to  be  done  during  the  re-reading. 
At  this  time  as  a  rule  the  numerals  only  are  included  in  the  reading, 
whereas  during  the  first  reading  all  of  the  contextual  elements  of  the 
problem,  both  words  and  numerals,  are  included. 

4.     SUMMARY 

The  following  conclusions  may  be  drawn  from  this  study:  (i)  The 
objects  of  the  re-readings  from  the  problems  were  almost  invariably 
numerals.  (2)  The  numerals  were  re-read  for  copying  on  the  computation 
sheets.  (3)  One  re-reading  for  copying  was  sufficient  for  most  of  the 
numerals.  Some  of  the  longer  numerals,  however,  were  re-read  in  two 
parts.  (4)  The  numerals  of  greater  length  required  longer  times  for  re- 
reading, on  the  part  of  a  majority  of  the  subjects.  (5)  The  total  time 
required  for  re-reading  the  numerals  was  less  than  the  time  required 
for  the  first  reading,  with  three  of  the  four  subjects. 


CHAPTER  V 

READING  NUMERALS  IN  COLUMNS— FOURTH  PRELIMINARY  STUDY 
I.      DESCRIPTION  OF  THE   STUDY 

The  purpose  of  this  study  was  to  give  some  description  of  the  reading 
of  numerals,  when  numerals  only  appeared  as  the  material  to  be  read  and 
when  each  individual  numeral  was  placed  in  a  separate  line.  The 
general  plan  followed  in  procuring  the  data  was  to  have  adult  subjects 
copy  numerals  from  the  pages  on  which  they  appeared  onto  other  sheets, 
and  at  the  same  time  articulate  the  numerals  in  an  easy  natural  way. 
This  articulation  was  recorded  by  the  author  of  this  report.  By  means 
of  a  system  of  notes  which  will  be  described  later,  it  was  possible  to  get 
a  fairly  full  account  of  what  was  said  and,  as  experience  in  making  the 
records  accumulated,  it  was  possible  to  distinguish  clearly  the  various 
types  of  reading. 

The  subjects  were  four  graduate  students  in  the  School  of  Education 
of  the  University  of  Chicago.  Subjects  R,  L,  and  G  had  each  served  in 
the  second  preliminary  study.  Subjects  G  and  H  had  read  the  problems 
of  the  first  preliminary  study  and  photographic  records  of  the  eye- 
movements  of  both  H  and  G  appear  in  the  second  part  of  this  report. 
The  materials  which  they  read  were  forty-eight  ordinary  whole  numerals 
varying  in  length  from  one  to  seven  digits,  and  including  seven  numerals 
for  each  different  digit-length  except  that  there  were  only  six  numerals 
of  seven-digit  length.  Punctuation  in  the  form  of  commas  was  used  in 
the  customary  way  with  some  of  the  five-,  six-,  and  seven-digit  numerals; 
and  with  some  of  these  numerals  it  was  not  used.  Four  numerals  of 
both  the  five-  and  six-digit  lengths,  and  three  numerals  of  the  seven-digit 
length  were  punctuated,  while  three  numerals  of  each  of  the  five-,  six-, 
and  seven-digit  lengths  were  not  punctuated.  The  numerals  were  type- 
written on  separate  lines  and  so  arranged  that  the  tens,  hundreds,  etc., 
places  of  the  numeral  above  were  not  exactly  above  the  same  places  of 
the  numeral  in  the  line  below.  Subjects  R  and  G  each  read  the  whole 
set  of  numerals  twice,  while  the  other  two  subjects  read  each  set  only 
once. 

The  subjects  copied  the  numerals  from  the  text  sheet  on  to  the  copy 
sheet  at  normal  speed.  At  the  same  time  they  articulated  the  numerals 
in  an  easy  low  voice  which  the  observer  was  able  to  hear  at  a  distance  of 

24 


READING  NUMERALS  IN  COLUMNS  25 

approximately  two  feet.  That  part  of  the  instructions  which  called  for 
copying  the  numerals  was  inserted  in  order  to  provide  a  genuine  working 
purpose  for  reading  them.  At  the  same  time  this  purpose  required  an 
exact  reading  of  every  numeral.  The  kind  of  reading  done,  therefore, 
in  compliance  with  these  instructions  was  of  a  relatively  clearly  denned 
functional  type,  quite  similar  in  obvious  ways  to  the  re-reading  of 
numerals  for  copying,  which  was  described  in  the  preceding  study.  The 
provision  for  articulation  enabled  the  observer  to  report  the  readings  in 
so  far  as  the  grouping  of  the  digits  of  the  numerals  and  the  numerical 
language  used  in  reading  them  were  concerned.  At  the  same  time  the 
articulation  did  not  seem  to  interfere  with  the  reading. 

Numerals  are  said  to  be  read  by  digit  groups  when  certain  successive 
digits  are  so  closely  associated  with  each  other  in  the  reading  as  to  form 
units  of  reading,  which  units  are  at  the  same  time  clearly  distinguished 
from  other  similar  units.  The  digits  which  constitute  a  group  are  bound 
together  by  being  pronounced  in  quick  succession  as  one  series.  The 
pronunciations  of  the  several  digit  groups  are  separated  from  each  other 
by  tune  intervals  distinctly  longer  than  the  time  intervals  which  separate 
the  pronunciations  of  the  individual  digits. 

For  reporting  the  digit  groups  and  the  numerical  language  used  in 
reading  the  numerals  a  simple  code  system  was  devised  which  was 
based  on  the  symbols  i,  2,  and  3  signifying  respectively  the  grouping 
of  the  digits  of  a  numeral  in  groups  of  one,  two,  and  three  digits.  A  few 
other  signs  were  necessarily  added  to  indicate  various  modifications  of 
these  groups.  A  brief  period  of  training  with  sets  of  practice  numerals 
was  undergone  by  the  observer  and  by  each  of  the  subjects.  After  this 
training,  they  were  able  to  proceed  with  the  experiment  in  full  conformity 
with  the  instructions.  The  data  which  were  obtained  during  the  course 
of  this  study  are  arranged  in  tables  XI-XV. 

2.       GENERAL    DESCRIPTION   OF   THE   THREE   MAIN   GROUPS   USED 

The  most  striking  feature  of  the  reading  of  numerals,  which  was 
discovered  in  this  study,  was  the  fact  that  the  subjects  habitually 
divided  the  numerals  into  digit  groups.  Three  different  sizes  of  groups 
were  clearly  distinguished  in  the  readings,  namely,  those  that  were  made 
up  of  one,  two,  and  three  digits  respectively.  The  one-digit  groups 
appeared  more  frequently  in  the  one-,  three-,  and  seven-digit  numerals 
as  is  shown  in  Table  XII.  The  two-  and  three-digit  groups  appeared 
in  the  readings  of  numerals  of  all  the  greater  digit-lengths.  Relatively 
large  numbers  of  three-digit  groups  appear  in  readings  of  the  five-,  six-, 


26 


HOW  NUMERALS  ARE  READ 


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5 
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C/2 

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X 

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READING  NUMERALS  IN  COLUMNS  27 

and  seven-digit  numerals.  In  the  cases  of  the  six-  and  seven-digit  nu- 
merals the  explanation  of  this  condition  lies  in  the  fact  that  numerals  of 
these  lengths  present  just  twice  as  many  opportunities  for  the  employ- 
ment of  three-digit  groups  as  numerals  of  four-  or  five-digit  lengths.  In 
the  case  of  the  five-digit  numerals,  on  the  other  hand,  the  large  number 
of  three-digit  groups  is  attributable  to  the  remarkable  uniformity  with 
which  the  habit  of  reading  five-digit  numerals  in  two  groups  of  two  and 
three  digits  respectively  was  followed  by  all  of  the  subjects. 

The  three-digit  groups  exhibited  two  distinct  types  which  are 
referred  to  as  the  simple  and  complex  types.  The  three  digits  of  the 
simple  type  of  three-digit  groups  are  pronounced  individually  and  with 
equal  tune  intervals  between  them.  The  three  digits  of  the  complex 
type  on  the  other  hand  are  pronounced  in  two  distinct  subgroups,  the 
first  of  which  includes  one  digit  and  the  second,  two  digits.  Both  types 
appear  in  the  readings  of  the  five-,  six-,  and  seven-digit  numerals,  as 
is  seen  in  the  readings  of  Subject  G,  which  are  reported  in  detail  in 
Table  XIV. 

Habit  on  the  part  of  the  individual  subject  appears  as  the  most 
conspicuous  factor  in  determining  which  of  the  two  types  of  three-digit 
group  was  chosen,  when  a  three-digit  group  was  used.  The  last  right- 
hand  column  of  Table  XIII  discloses  the  fact  that  Subject  H  used  the 
simple  type  of  three-digit  group  only,  while  Subject  G  used  it  in  a 
preponderant  number  of  cases.  On  the  other  hand  Subject  R  used  the 
complex  type,  almost  invariably. 

3.      MAIN-GROUP   PATTERNS   FOR   NUMERALS   OF   LIKE    LENGTH 

It  became  apparent  early  in  the  course  of  this  study  that  the  digits 
of  numerals  of  any  one  length  were  being  grouped  in  much  the  same  way 
by  all  of  the  subjects.  This  observation  is  strikingly  confirmed  by 
the  data  which  are  presented  in  Table  XI.  The  fact  that  appears  most 
strikingly  after  an  examination  of  this  table  is  that  the  digits  of  numerals 
of  any  particular  length  are  divided  into  a  certain  number  of  groups, 
which  groups  are  made  up  of  certain  numbers  of  digits  and  stand  in  a  cer- 
tain order  of  succession.  The  case  of  the  seven-digit  punctuated  numerals 
will  serve  for  illustration.  The  digits  of  these  numerals  are  seen  to  have 
been  divided  into  three  groups  by  all  of  the  subjects.  In  the  first  group, 
one  digit  is  found  almost  invariably,  in  the  second  group  three  digits, 
and  in  the  third  group  three  digits.  To  this  succession  of  digit  groups 
of  certain  sizes  only  one  exception  occurs.  Such  an  arrangement  of  the 
digits  of  a  numeral  is  designated  as  a  main-group  pattern. 


28 


HOW  NUMERALS  ARE  READ 


The  one-  and  two-digit  numerals  were  each  read  as  single  groups  of 
one  and  two  digits  respectively.  The  first  variation  from  one  main- 
group  pattern  as  representative  of  the  reading  of  numerals  of  the  same 
length  occurs  in  the  three-digit  numerals,  which  exhibit  two  patterns. 

TABLE  XII 

NUMBER  OF  ONE-,  Two-,  AND  THREE-DIGIT  GROUPS  USED  IN  READING  NUMERALS 
OF  THE  SEVERAL  DIGIT-LENGTHS  IN  COLUMNS 


Digit  -Length  of  Numerals 

i 

2 

3 

4 

5 

6 

7t 

Number  of  readings  given  numerals  . 
Number  of: 

42 
42 

42 

42 

25 
25 
17 

36* 

2 

68 

2 

42 

3 

45 
30 

42 

28 
65 

36 

35 
30 
5i 

Two-digit  groups 

42 

*  There  are  36  readings  of  four-digit  numerals  when  the  6  readings  of  the  numeral  1000,  which 
was  always  read  as  "one  thousand,"  are  omitted. 

t  One  four-digit  group  was  used  by  Subject  L  in  reading  one  of  the  seven-digit  numerals. 

TABLE  XIII 

NUMBER  OF  SIMPLE  (3)  AND  OF  COMPLEX  (1-2)  THREE-DIGIT  GROUPS  USED  IN 
READING  FIVE-,  Six-,  AND  SEVEN-DIGIT  NUMERALS  IN  COLUMNS 


DIGIT-LENGTH  OF  NUMERALS 


5 

6 

7 

5-7 

5-7 

Punc- 
tuated 

Non- 
Punc- 
tuated 

Punc- 
tuated 

Non- 
Punc- 
tuated 

Punc- 
tuated 

Non- 
Punc- 
tuated 

Punc- 
tuated 

Non- 
Punc- 
tuated 

Punc- 
tuated 
and 
Non- 
Punc- 
tuated 

Subject  R— 
Simple  three-digit  groups  

5 
ii 

13 
3 

8 

12 

2 

7  " 
5 

6 
5 

6 

10 
3 

2 

12 

24 

25 
10 

18 

"28" 

7 
3 

4 

12 

52 

32 
13 

22 

Complex  three-digit  groups  

8 
6 

2 

4 

6 

2 

3 

2 

Subject  G  — 
Simple  three-digit  groups  
Complex  three-digit  groups  

Simple  three-digit  groups  
Complex  three-digit  groups 



Subject  L  — 
Simple  three-digit  groups  .... 

I 
3 

I 

5 
3 

3 

2 

3 

2 

8 
9 

6 

i 

14 
10 

Complex  three-digit  groups  

The  four-digit  numerals  appear  almost  invariably  in  a  pattern  of  two 
groups  of  two  digits  each.  A  conspicuous  exception  to  this  regular  main- 
group  pattern  of  the  four-digit  numerals  is  found  in  the  familiar  numeral 
1000  which  was  regularly  read  as  "one  thousand."  This  exception  is 


READING  NUMERALS  IN  COLUMNS  29 

further  evidence  of  the  fact,  to  which  attention  has  been  called  in  other 
sections  of  this  report,  that  this  numeral  is  different  in  quality  from  other 
numerals  of  the  same  length. 

The  five-digit  numerals  show  in  a  preponderant  number  of  cases  a 
pattern  of  two  groups  of  two  and  three  digits  respectively.  The  domi- 
nant pattern  for  the  six-digit  numerals  is  that  of  two  groups  of  three 
digits  each.  The  first  main-group  patterns  made  up  of  three  groups  to 
appear  among  punctuated  numerals  are  found  in  the  seven-digit  numer- 
als. When  numerals  of  any  of  the  greater  digit-lengths  are  written 
without  punctuation,  other  patterns  than  the  main-group  pattern  for 
that  digit-length  make  their  appearance.  In  the  case  of  non-punctuated, 
seven-digit  numerals  as  many  as  ten  different  main-group  patterns 
were  found.  With  all  of  the  other  digit-lengths,  however,  conformity  to 
main-group  pattern  is  clearly  the  rule  with  all  subjects.  Evidently  such 
arrangements  of  the  digits  of  numerals,  when  they  are  being  read  for  copy- 
ing, are  procedures  which  have  been  very  thoroughly  conventionalized 
by  long  practice,  or  else  such  procedures  rest  closely  upon  certain  funda- 
mental laws  of  mental  action. 

4.      VARIATIONS   IN   NUMERICAL   LANGUAGE 

The  fact  that  all  numerals  of  a  certain  digit-length  were  usually 
read  in  the  same  main-group  pattern  does  not  mean  that  the  language 
used  in  reading  them  was  the  same.  Variations  in  language  were  found 
in  the  readings  of  both  two-  and  three-digit  groups.  By  the  use  of 
symbols  to  represent  each  of  these  variations,  the  detailed  numerical 
language  which  was  used  by  one  subject,  namely  by  Subject  G,  is  given 
in  Table  XIV  for  the  readings  of  all  the  numerals.  The  patterns  which 
appear  in  this  table  may,  therefore,  be  designated  as  numerical-language 
patterns. 

Four  different  numerical-language  patterns  are  found  in  the  readings 
by  Subject  G  of  the  punctuated  numerals  of  five-digit  lengths.  Any 
five-digit  numeral  may  be  pronounced  according  to  each  of  these 
patterns.  The  numeral  76,184,  which  was  one  of  the  numerals  read  by 
•the  subjects,  may  be  taken  as  an  example  of  the  five-digit  numerals. 
When  this  numeral  is  pronounced  successively  according  to  each  of  the 
four  numerical-language  patterns,  as  they  are  represented  from  top  to 
bottom  in  the  column  of  punctuated  five-digit  numerals,  the  following 
four  different  pronunciations  result:  " seventy-six — one  eight  four"; 
" seven  six — one,  eight  four";  " seven  six — one  eight  four";  "seventy- 
six — one,  eighty-four." 


HOW  NUMERALS  ARE  READ 


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READING  NUMERALS  IN  COLUMNS  31 

A  second  examination  of  the  four  numerical-language  patterns,  the 
pronunciations  of  which  are  presented  immediately  above,  reveals  the 
fact  that  all  four  of  the  patterns  are  modifications  of  one  fundamental 
main-group  pattern.  This  fundamental  pattern  contains  two  groups, 
the  first  of  which  is  a  two-digit  group,  while  the  second  is  a  three-digit 
group.  It  is  the  variations  that  appear  in  the  pronunciations  of  both 
of  these  groups  that  distinguish  the  four  different  numerical-language 
patterns.  In  the  two-digit  groups  the  differences  in  language  are 
merely  those  between  the  words,  "seven  six"  and  " seventy-six, "  or 
again  between  "eight  four"  and  "eighty-four."  In  the  three-digit 
groups  the  words  used  to  pronounce  the  simple  type  differ  from  those 
used  to  pronounce  the  complex  type,  as  "one  eight  four"  differs  from 
"one,  eighty-four." 

The  habits  of  individual  subjects  were  discovered  in  a  previous 
paragraph  to  be  the  most  conspicuous  factors  in  determining  which  of 
the  two  types  of  three-digit  groups  was  used.  The  original  records  show 
that  such  individual  habits  similarly  were  the  chief  factors  in  determining 
what  language  was  used  in  pronouncing  the  two-digit  groups.  On  the 
other  hand,  the  fundamental  main-group  pattern  for  any  length  of  numeral 
appeared  consistently  in  the  readings  of  all  subjects.  It  is,  therefore, 
apparent  that  the  selection  of  the  digit-length  of  groups  and  the  selection 
of  the  order  of  their  appearance  are  more  fundamental  phases  of  the 
reading  of  the  numerals  than  the  selection  of  the  particular  type  of 
three-digit  group  and  the  choice  of  the  particular  words  by  which  the 
two-  and  three-digit  groups  are  to  be  pronounced. 

5.      INFLUENCE    OP   PUNCTUATION   ON   THE    GROUPING   OF   DIGITS    OF 
LONGER   NUMERALS 

Great  differences  appear  between  the  readings  of  punctuated  and 
non-punctuated  numerals  in  the  number  of  both  two-  and  three-digit 
groups  which  are  employed.  Punctuation  apparently  has  the  effect  of 
increasing  the  number  of  three-digit  groups  used,  and  conversely  of 
decreasing  the  number  of  two-digit  groups.  In  the  extreme  right-hand 
column  of  Table  XV  it  is  seen  that  each  of  the  last  three  subjects  used 
a  much  greater  proportion  of  three-digit  groups  for  the  punctuated 
numerals,  and  on  the  other  hand  a  much  greater  proportion  of  two-digit 
groups  among  the  non-punctuated  numerals. 

The  preponderance  of  three-digit  groups  in  the  punctuated  numerals 
and  conversely  the  preponderance  of  two-digit  groups  in  the  non- 
punctuated  numerals  are  each  relatively  much  greater  for  the  six-  and 


32 


HOW  NUMERALS  ARE  READ 


seven-digit  numerals  than  for  five-digit  numerals.  Such  a  situation  may 
be  partly  explained  by  the  fact  to  which  attention  was  called  above, 
that  it  is  possible  to  use  just  twice  as  many  three-digit  groups  in  reading 
a  six-  or  seven-digit  numeral  as  in  reading  a  five-digit  numeral.  Fewer 
main-group  patterns  appear  in  the  columns  for  punctuated  numerals  in 
Table  XI  than  in  the  columns  for  non-punctuated  numerals.  Examina- 
tion of  the  patterns  in  both  columns  shows  that  there  are  greater 
numbers  of  groups  in  the  non-punctuated  patterns,  and  this  is  due 
mainly  to  the  more  frequent  use  of  the  smaller  group  of  two  digits. 

The  readings  of  one  subject,  R,  exhibited  practically  no  differences 
in  selection  of  two-  and  three-digit  groups,  which  may  be  attributed  to 

TABLE  XV 

EFFECT  OF  PUNCTUATION  ON  THE  NUMBER  OF  Two-  AND  THREE-DIGIT  GROUPS  USED 
IN  READING  FIVE-,  Six-,  AND  SEVEN-DIGIT  NUMERALS  IN  COLUMNS 


DIGIT-LENGTH  OF  NUMERALS 

5 

6 

7 

5-7 

Punc- 
tuated 

Non- 
Punc- 
tuated 

Punc- 
tuated 

Non- 
Punc- 
tuated 

Punc- 
tuated 

Non- 
Punc- 
tuated 

Punc- 
tuated 

Non- 
Punc- 
tuated 

Subject  R— 
Two-digit  groups  
Three-digit  groups  
Subject  G  — 
Two-digit  groups  
Three-digit  groups  
Subject  H— 
Two-digit  groups  
Three-digit  groups  
Subject  L— 
Two-digit  groups 

8 
8 

8 
8 

4 
4 

4 
4 

6 
6 

7 

5 

4 

2 

4 

2 

3 

10 
12 

4 
8 

2 

4 

2 

8 
36 

10 

35 

4 
18 

5 
17 

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34 
II 

21 

4 

12 

7 

16 

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........ 

...... 

12 

15 
2 

9 

4 
3 

12 

2 
II 

6 

I 

5 

Three-digit  groups  

punctuation.  With  the  larger  numerals  he  used  three-digit  groups 
consistently,  wherever  it  was  possible  to  use  them,  in  both  punctuated 
and  non-punctuated  numerals.  The  exception  in  this  case  is  probably 
attributable  to  his  having  attained  to  a  relatively  high  stage  of  proficiency 
in  the  mechanical  processes  of  reading  numerals  by  means  of  a  large 
amount  of  special  practice  in  a  kind  of  reading  of  numerals  which  is  very 
similar  to  that  used  in  this  study.  This  practice  he  had  gained  while 
earning  his  living  in  the  capacity  of  railroad  rate  clerk.  A  large  part  of 
his  work  was  to  read  numerals  and  call  them  off  to  a  colleague,  who 
copied  them  on  other  paper. 

The  easy  use  of  larger  digit  groups  seems  to  give  greater  facility  and 
greater  speed  to  the  reading  of  numerals.     The  value  of  punctuation 


READING  NUMERALS  IN  COLUMNS  33 

to  the  subjects  in  large  part  lies  in  the  fact  that  its  employment  encour- 
aged the  use  of  the  larger  group  of  three  digits.  The  subject  confronted 
with  the  necessity  of  reading  a  large  but  unknown  number  of  unspaced 
digits  is  in  a  difficult  situation.  Such  situations  are  not  frequently  encoun- 
tered in  the  experiences  of  the  ordinary  reader.  In  consequence  he  pro- 
ceeds with  caution  and  with  the  smaller  groups  of  one  and  two  digits. 
The  great  number  of  small  groups  and  the  large  number  of  group  patterns 
which  were  employed  in  the  readings  of  non-punctuated  numerals  of 
seven-digits  length,  as  shown  in  Table  XI,  are  evidently  results  of 
procedure  under  difficulty  and  with  uncertainty.  The  same  situation 
produced  the  great  number  and  variety  of  numerical-language  patterns 
in  the  readings  by  Subject  G  of  the  same  numerals.  In  situations  such 
as  these,  employment  of  the  symbols  of  punctuation  appears  to  afford 
great  and  immediate  relief. 

6.      PERSISTENCE    OF    PATTERNS    FROM    THE    FIRST    READING    THROUGH 
A    SECOND    READING 

.  Opportunity  to  study  the  persistence  of  the  main-group  and  numerical- 
language  patterns,  which  were  found  in  the  first  reading  of  the  numerals, 
through  the  second  reading  of  the  same  numerals  was  given  in  the  cases 
of  two  subjects.  Subjects  R  and  G  each  read  all  of  the  numerals  at  two 
separate  readings.  The  interval  of  time  between  the  two  readings  was 
approximately  30  minutes  with  each  subject.  It  was  found  that  both 
R  and  G  read  the  same  numerals  in  the  same  patterns  at  both  readings 
with  very  few  exceptions.  Subject  R,  who  was  more  highly  trained  in 
the  reading  of  numerals  than  any  other  subject,  made  fewer  changes 
than  G.  More  changes  were  made  in  numerical-language  patterns  than 
in  main-group  patterns.  Most  of  the  changes,  which  were  made,  were 
found  in  the  non-punctuated  numerals  of  seven-digits  length. 

The  fact  that  the  same  main-group  patterns  so  consistently 
reappeared  at  the  different  readings  of  these  subjects  gives  further 
evidence  in  support  of  the  conclusion,  which  was  advanced  in  a  paragraph 
above,  that  the  arrangement  of  digits  in  main-group  patterns  has  been 
very  thoroughly  conventionalized,  or  else  that  such  procedure  rests 
closely  upon  certain  fundamental  laws  of  mental  action. 

7.      SUMMARY   OF   CONCLUSIONS 

The  following  conclusions  are  drawn  from  the  data  presented  in 
this  study  concerning  the  articulated  reading  of  numerals  for  copying, 
(i)  The  digits  of  numerals  are  grouped  in  the  process  of  reading.  The 


34  HOW  NUMERALS  ARE  READ 

groups  of  digits  are  of  three  sizes,  namely,  of  one,  two,  and  three  digits 
respectively.  (2)  The  numerals  of  each  of  several  digit-lengths  are  read 
almost  invariably  in  a  main-group  pattern  which  is  peculiar  to  that 
digit-length.  (3)  Various  numerical-language  patterns  are  used  in 
pronouncing  numerals  of  the  same  length.  (4)  The  employment  of 
punctuation  with  the  longer  numerals  encourages  the  use  of  three-digit 
groups  and,  conversely,  discourages  the  use  of  two-digit  groups  in  the 
reading.  A  larger  group  unit  is  thus  secured.  (5)  The  main-group  and 
numerical-language  patterns  which  are  used  in  the  first  reading  of 
numerals  persist  for  the  most  part  through  a  second  reading  of  the 
same  numerals. 


PART  II.     STUDIES  OF  THE  READING  OF  NUMERALS— 
BY  USE  OF  PHOTOGRAPHIC  APPARATUS 

CHAPTER  VI 

DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES 
I.      APPARATUS  DESCRIBED 

The  data  which  are  presented  in  the  remaining  sections  of  this 
report  were  obtained  through  the  use  of  an  apparatus  designed  to  record 
the  movements  of  the  eyes  in  reading  by  means  of  photography.  The 
apparatus  is  described  and  its  use  explained  in  a  monograph  by  Dr.  C.  T. 
Gray,1  and  excellent  photographs  and  diagrams  of  the  same  are  found 
in  a  magazine  article  by  Gilliland.2  A  few  slight  adaptations  of  the 
apparatus  and  procedure,  which  are  described  in  these  references,  were 
necessary  in  view  of  the  materials  and  purposes  of  the  present  investiga- 
tion. The  materials  which  were  read  by  the  subjects  of  this  investigation 
were  printed  on  separate  cards  eight  and  one-half  by  four  inches  in  size. 
These  cards  were  placed  on  the  stand  immediately  before  the  lenses  of 
the  camera  and  directly  before  the  eyes  of  the  readers.  A  flood  of  light 
reflected  from  the  overhead  mirror  gave  bright  illumination  to  any 
materials  which  were  placed  upon  the  stand.  A  convenient  elbow-rest 
was  provided  for  the  right  arm  in  such  a  manner  that  computation  with 
a  pencil  could  be  undertaken  easily  and  comfortably,  and  directly  upon 
the  problem  card,  whenever  the  subject  chose  to  do  so.  With  this 
arrangement  it  was  possible  for  the  pencil  of  light  which  is  reflected 
from  the  eye,  to  register  continuously  upon  the  film  during  periods  of 
computation,  as  well  as  during  periods  of  reading  from  the  problem. 

2.      THREE   TYPES   OF   READING-MATERIALS   USED 

The  reading-materials  which  were  selected  for  this  part  of  the 
investigation  were  of  three  different  types,  namely,  simple  arithmetical 
problems,  numerals  isolated  in  lines,  and  a  paragraph  of  ordinary 
expository  prose. 

1  C.  T.  Gray,  "Types  of  Reading  Ability  as  Exhibited  Through  Tests  and  Labora- 
tory Experiments,"  Supplementary  Educational  Monographs,  Vol.  I,  No.  5  (1917), 
pp.  83-91. 

2  A.  R.  Gilliland,  "Photographic  Methods  for  Studying  Reading,"  Visual  Educa- 
tion, Vol.  II,  No.  2  (February,  1921),  pp.  21-26. 

35 


36  HOW  NUMERALS  ARE  READ 

The  arithmetical  problems  were  so  designed  as  to  provide  a  simple 
and  genuine  problem-setting  for  the  numerals  which  had  been  selected 
for  further  study.  The  numerals  thus  selected  included  representatives 
from  each  of  the  several  lengths  of  from  one  to  seven  digits,  the  familiar 
numeral  1000,  and  a  group  of  three  numerals  placed  closely  together  in 
one  problem.  Further  details  concerning  the  problems  are  given  in 
Table  XVI  and  the  problems  exactly  as  they  were  read  by  the  subjects 
appear  as  Selection  i. 

TABLE  XVI 

DESCRIPTION  OF  THE  FIVE  PROBLEMS  READ  IN  THE  PHOTOGRAPHIC  APPARATUS 


NUMBER  OF 
PROBLEM 

ORDINAL 

NUMBER 
OF  EACH 
LINE  IN 
PROBLEM 

LENGTH 

OF 

LINE  IN 
MILLI- 
METERS 

THE  NUMBER  IN  EACH  LINE  OF 

Words 

Letters 

Numerals 

Digits 

Words 
and 
Numerals 

Letters 
and 
Digits 

i  

ISt 

fist 

\2d 

fist 

2d 

U 
fist 

2d 

l3d 

fist 
J2d 
Ud 

76 

y 

95 

IO2 

86 
95 

102 
90 

95 

102 

94 

9 

ii 

8 

ii 

7 
9 

10 

8 
8 

8 

10 
12 

34 

4° 
37 

44 
34 
37 

42 
56 
30 

40 
43 
45 

2 

i 
i 

0 
2 
0 

I 

o 
3 

i 
i 
o 

3 

3 

4 

0 

ii 
o 

4 

0 

9 

7 
6 

0 

ii 

12 

9 

II 
9 
9 

II 
8 
ii 

9 
II 

12 

37 

43 
4i 

44 
45 
37 

46 
56 
39 

47 
49 
45 

2  

3             •    • 

4        

5  

Total  for  all 
problems.  . 

Average  num- 
ber per  line 

12 

III 

482 

12 

47 

123 

529 

93-33 

NOTE. — The  number  of  spaces  between  words  is  not  counted. 


SELECTION  1 
FIVE  PROBLEMS  READ  BEFORE  PHOTOGRAPHIC  APPARATUS 

1.  At  47  cents  a  dozen  what  will  2  dozen  eggs  cost  ? 

2.  A  timber  man  owns  one  plot  of  357  acres,  and  another  of 
1643  acres.     How  much  ground  does  he  own  altogether  ? 

3.  A  wholesale  grain  firm  had  at  the  beginning  of   the  day 
243,987  bushels  of  wheat.     During  the  day's   trading  21,765 
bushels  were  sold.     How  much  did  they  then  have  ? 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES  37 

4.  A  commission  house  had  on  hand  1000  cases  of  canned  corn. 
From  three  different  canning  factories  they  bought  respectively 
1276,  91,  and  817  cases.     How  many  did  they  then  have? 

5.  If  one  telephone  company  uses  1,918,564  cross  bars  during 
the  year,  and  another  company  in  the  same  period  uses  617,453 
cross  bars,  how  many  more  does  the  one  use  than  the  other  ? 

The  numerals  isolated  in  lines  included  a  list  of  thirty-four  numerals. 
Twenty-eight  of  the  list  of  thirty-four  consisted  of  ordinary  numerals 
which  were  selected  by  taking  four  numerals  from  each  of  the  seven- 
numeral  lengths  of  one  to  seven  digits.  In  addition,  the  list  included  the 
six  special  form  numerals,  namely,  1000,  333,  25,000,  o,  99,  and  637,637. 
They  were  presented  to  the  readers  on  two  different  cards.  The  line 
space  between  any  two  numerals  was  16  mm.,  and  the  lines  were  placed 
two  spaces  apart.  These  two  details  of  arrangement  were  followed  in 
order  that  the  reading  of  any  numeral  might  be  entirely  separate  from 
the  reading  of  any  other.  The  numerals  are  reproduced  as  Selection  2 
and  in  the  same  form  in  which  they  were  presented  to  the  readers. 

SELECTION  2 
ISOLATED  NUMERALS  READ  BEFORE  PHOTOGRAPHIC  APPARATUS 


(Card  One) 
836  3  5489  756,352  46  4,325,986 

85,974  239  1  16,789  1024  354,908 

12  2,374,957  1000  333  25,000 


(Card  Two) 

76,184                 9317                  17                  2  5,236,795                 256 

743,819                  1928                365                  8  93,548                  3,984,673 

107,308                  52                  0                  99  637,637 


38  HOW  NUMERALS  ARE  READ 

The  selection  of  ordinary  expository  prose  was  taken  from  Judd's 
Psychology  of  High-School  Subjects.  The  subjects  read  directly  from  the 
book.  Data  were  tabulated  from  the  reading  of  ten  lines  by  each  subject- 
Because  of  defects  in  the  records  of  subjects  W  and  G  only  five  and  seven 
lines  respectively  were  tabulated  from  their  readings.  The  regularity 
in  number  and  duration  of  pauses  found  in  the  data  for-  the  few  lines, 
which  were  tabulated  for  these  subjects,  however,  give  evidence  that  the 
data  for  these  few  lines  represent  the  normal  reading  of  these  subjects 
in  these  materials.  The  record  which  represented  the  reading  of  Hb 
was  totally  unsatisfactory  for  use.  The  ten  lines  of  the  text  were  each 
93  mm.  in  length  and  included  101  words  and  452  letters.  They  are 
reproduced  as  Selection  3. 

SELECTION  3 
ORDINARY  PROSE  READ  BEFORE  PHOTOGRAPHIC  APPARATUS 

"Anyone  who  has  struggled  with  the  German  language  has  an 
appreciation  of  the  satisfaction  which  the  novice  feels  in  watching 
the  way  an  expert  in  this  language  manages  a  separable  verb.  The 
moment  the  verb  is  used  in  a  sentence,  there  arises  a  feeling  of  craving 
for  the  remainder  of  the  verb.  The  skillful  German  places  between  the 
verb  and  the  prefix  a  long  series  of  phrases  and  words,  but  ultimately 
arrives  with  perfect  precision  at  the  end  of  the  sentence,  and  gives  the 
satisfaction  which  comes  from  a  proper  closing  of  the  feeling  which  was 
started  when  the "x 

3.      INSTRUCTIONS  TO   SUBJECTS   AND   DESCRIPTION   OF   SUBJECTS 

The  instructions  given  the  readers  were  varied  for  each  of  the  three 
kinds  of  materials  which  were  read.  When  the  problems  were  being 
read  the  subjects  were  asked  to  attack  them  with  the  normal  problem- 
solving  attitude.  Each  individual  was  provided  with  a  pencil  which  he 
was  told  he  might,  or  might  not,  use  in  computation,  as  he  chose.  The 
instructions  which  the  readers  received  for  the  isolated  numerals  were 
to  read  the  numerals  successively  in  the  lines,  to  read  all  of  them  accu- 
rately, and  to  proceed  at  the  normal  rate  of  speed.  Each  numeral  was 
to  be  articulated  in  an  easy,  natural  manner  and  in  a  voice  which  was 
barely  audible  to  the  observer,  who  stood  at  a  distance  of  approximately 
three  feet.  Provision  for  this  slight  articulation  was  included  as  a  means 
of  encouraging  the  complete  reading  of  all  numerals.  For  the  expository 
prose  selection  the  instructions  were  to  read  silently  for  a  clear  under- 

1  C.  H.  Judd,  Psychology  of  High-School  Subjects.  Boston:  Ginn  &  Co.,  1915. 
Pp.  190. 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES  39 

standing  of  the  paragraph  and  at  normal  speed.  The  volume  from  which 
the  selection  was  taken  was  familiar  in  a  general  way  to  all  of  the  readers. 
They  were  given  its  title  in  advance  and  the  subject-matter  of  the  passage 
to  be  read  was  described  as  relating  to  the  psychology  of  language. 

The  six  subjects,  records  of  whose  readings  appear  in  this  part  of  the 
investigation,  were  all  male  graduate  students  in  the  School  of  Education 
of  the  University  of  Chicago.  Three  of  them  had  served  in  various 
preliminary  studies.  Subject  G  had  read  the  problems  of  the  first  and 
second  preliminary  studies  and  was  classified  as  a  pronounced  partial 
first  reader  of  numerals  in  problems.  Subjects  H  and  Hb  had  served 
in  the  first  and  second  studies  respectively  and  were  both  found  to  be 
pronounced  whole  first  readers.  The  original  plan  of  the  investigation 
specified  that  the  subjects  whose  types  of  reading  had  been  studied  in 
the  preliminary  sections  should  act  as  subjects  for  the  eye-movement 
experiments.  Of  the  photographic  records  which  were  made  of  the 
subjects  of  previous  studies,  however,  only  those  of  G,  H,  and  Hb  were 
entirely  satisfactory.  None  of  the  subjects  reported  past  experiences 
which  seemed  likely  to  have  had  important  influence  upon  his  reading 
of  the  materials  of  this  study. 

4.   PROCEDURE  ON  THE  PART  OF  THE  OBSERVER 

The  instructions  were  given  to  the  subjects  before  they  took  their 
seats  at  the  camera,  and  samples  of  each  of  the  three  kinds  of  materials, 
which  were  to  be  read,  were  examined  by  them.  When  the  readers 
were  properly  seated  before  the  camera  they  were  given  a  brief  training 
with  practice  problems  and  with  practice  sets  of  isolated  numerals  until 
procedure  according  to  the  instructions  was  mastered.  After  the 
solving  of  each  problem,  several  of  the  subjects  were  asked  to  make 
brief  introspective  observations  concerning  whole  and  partial  first  reading 
of  numerals,  the  re-reading  of  numerals  and  the  steps  used  in  computa- 
tion. Their  reports  were  recorded  and  later  served  as  a  basis  for  inter- 
pretation of  the  corresponding  eye-movement  records. 

5.   GUIDE  FOR  READING  THE  PLATES 

The  photographic  films,  upon  which  the  lines  of  dots  representing 
pauses  of  the  eyes  were  recorded,  were  used  as  slides  in  a  projection 
lantern.  The  records  of  the  photographs  were  in  this  manner  projected 
upon  a  screen,  which,  at  the  same  time,  held  the  texts  of  the  various 
reading  materials.  The  photographic  picture  of  the  subject's  reading 
was  thus  superimposed  upon  the  exact  text  of  the  materials  which  he  had 
read.  It  was,  therefore,  possible  to  locate  directly  upon  the  lines  of  the 


40  HOW  NUMERALS  ARE  READ 

text  itself  the  letter  or  digit  about  which  the  attention  of  the  subject 
was  centered  at  any  pause  of  the  eye.  By  counting  the  number  of  dots 
in  the  lines  of  dots,  which  represented  the  pauses,  the  exact  durations 
of  the  pauses  were  ascertained. 

The  plates,  which  describe  readings  of  the  problems,  are  numbered 
I-XV  as  presented  in  this  chapter.  The  remaining  plates,  which 
describe  readings  of  the  isolated  numerals,  are  numbered  XVI-XXV  and 
are  found  in  chapter  ix.  The  lines  of  reading  materials  which  are  found 
in  the  plates  are  reproductions  of  the  lines  which  were  read  by  the 
subjects.  The  short  straight  vertical  lines  which  cross  the  lines  of  print 
represent  pauses  of  the  eye.  The  particular  letter,  digit  or  space  which 
is  crossed  by  a  vertical  line  represents  the  approximate  center  of  the  field 
of  perception  which  was  included  in  that  pause.  The  arabic  numbers, 
i,  2,  3,  etc.,  above  each  of  the  vertical  lines  indicate  the  serial  order  of 
each  pause  among  the  pauses  which  were  used  in  reading  the  problem. 
When  the  serial  number  of  a  pause  moves  to  the  left  of  the  serial  number 
of  the  previous  pause  a  backward  or  regressive  movement  of  the  eye  is 
indicated.  The  number  at  the  lower  end  of  a  pause  line  gives  the  dura- 
tion of  the  pause  in  units  of  1/50  of  a  second. 

The  vertical  lines  which  mark  the  pauses  used  in  the  first  reading  of 
the  problem  are  located  on  the  lines  of  the  reproduced  text.  All  pauses 
used  in  re-reading  or  in  copying  numerals  or  in  computation  are  recorded 
below  the  last  line  of  the  problem.  For  convenience  in  interpretation 
the  numerals  which  were  read  during  such  pauses  are  typewritten  below 
the  last  line  of  the  problem  and  directly  below  the  several  digit  spaces 
occupied  by  these  numerals  in  the  lines  below.  A  straight  horizontal 
line  below  the  vertical  lines  that  indicate  the  computation  pauses, 
describes  the  location  of  numerals  which  have  been  copied,  or  of  answers 
which  have  been  recorded. 

The  reading  of  the  plates  may  be  illustrated  by  the  reading  of 
Plate  XII,  which  is  as  follows:  Pause  i,  which  falls  in  the  first  line  of 
the  problem,  begins  the  first  reading  of  the  problem.  It  is  located  on  the 
letter  "a"  of  the  word  " wholesale"  and  the  duration  of  the  pause  is 
10/50  of  a  second.  Pause  2,  which  is  a  backward  or  regressive  movement 
from  Pause  i,  is  located  on  the  letter  "o"  of  "wholesale"  and  the 
duration  of  this  pause  is  also  10/50  of  a  second.  There  are  seven  pauses 
in  line  i.  Pause  9,  which  is  in  line  2,  falls  on  digit  "9"  of  the  numeral 
"  243,987"  and  its  duration  is  20/50  of  a  second.  It  is  a  regressive 
movement  from  Pause  8.  Pause  22  completes  the  first  readmg  of  the 
problem. 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES 


The  pauses  which  follow  represent  the  process  of  computation  and 
are  recorded  below  the  text  of  the  problem.  Pause  23,  which  is  the 
first  pause  used  in  the  process  of  computation,  is  apparently  a  locating 
pause.  It  is  followed  by  Pause  24,  by  which  digit  "7"  of  the  numeral 


z 
)ir|missi 


A  commission  ho 


ise  hati  on  hand  1 


of  canned  corn, 


i 

ning  factories  they  Bought  respectively 

/* 


IS    ffe    IT  IS      15         20          21 


22  23          24 

276,    l,    nd  $17  dases.     How  many  did  they  thei  ha\e? 


25 

ipoo 


First  reading  of  Problem  4  by  Subject  B  and  his  procedure  in  solving  the  problem. 
x  indicates  that  it  was  impossible  to  determine  with  precision  the  duration  of  the 
pause. 

"  243,987  "  is  read  in  41/50  of  a  second.  Attention  then  passes  immedi- 
ately with  Pause  25  to  digit  "5"  of  "21,765,"  which  digit  is  to  be 
subtracted  from  the  afore-mentioned  digit  "7."  Pauses  26,  27,  etc., 
continue  the  process  of  computation,  which  ends  with  Pause  32.  The 
subject  then  shut  his  eyes  and  the  record  was  finished. 

Plate  I  records  the  reading  of  Problem  4  by  Subject  B.  In  the  first 
reading  of  the  problem,  which  is  represented  by  the  pauses  in  the  lines 


42  HOW  NUMERALS  ARE  READ 

of  the  text  itself,  an  initial  regression  is  noted  in  line  i.  Pause  i  evidently 
was  not  located  closely  enough  to  the  left  end  of  the  line  for  a  satisfactory 
beginning.  Similar  initial  regressive  movements  appear  in  lines  2  and 
3.  During  pauses  1 6  to  19,  inclusive,  the  two  numerals  1276  and  91  were 
given  whole  first  readings.  With  Pause  24  the  first  reading  of  the 
problem  was  finished. 

PLATE  II 


irhissi< 


[        I 

A  comrhission  house  had  on  hahd  1 

18  12  10  II 


7    .«>  II  IZ  13  14. 


From  trjree  tiifTerent  banning  factories  they 


tespectr 


bought  respectively 


9    r  12  9  ro         13 


s.     JIow mp,ny  did  they  thfn  have? 

10  9 


iboo 


9 

First  reading  of  Problem  4  by  Subject  G  and  re-reading  the  numeral  1000 

The  remaining  pauses,  which  represent  subsequent  procedure  with 
the  problem,  are  placed  below  the  lines  of  the  text.  The  numeral  1000 
was  re-read  with  Pause  25.  The  records  do  not  give  sufficiently  detailed 
information  for  the  identification  of  the  purposes  of  the  individual  pauses 
subsequent  to  Pause  25.  The  computation  was,  however,  conducted 
directly  from  the  problem  card  and  apparently  the  subject  added  the 
numerals  1276  and  91  first,  and  then  added  817  to  that  result.  The 
answer  was  recorded  during,  or  immediately  after,  Pause  41. 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES  43 

Plate  II  contains  the  record  of  Subject  G  for  Problem  4.  He 
proceeded  rapidly  with  the  first  line,  but  read  the  second  line  with  a 
number  of  pauses  which  is  relatively  large  as  compared  with  the  number 
of  pauses  on  his  other  lines.  The  numeral  1000  was  read  in  detail  with 
Pause  4  in  a  time  interval  of  11/50  of  a  second.  With  pauses  15  and 
16  he  gave  partial  first  readings  to  the  three  numerals,  1276,  91,  and  817. 
Such  partial  first  readings  of  these  numerals  were  in  this  instance  suf- 
ficient preparation  for  computation  with  them.  The  first  reading  of 
the  problem  was  completed  with  Pause  21. 

PLATE  III 

IT    J  11  I  J    1 

At  4F  cents  d  doted  whafl  will  2  dozen  eggs  cost  ? 

It        I*.  35  7         10     19  «  i  24 


¥  f 

5ll4j3ZZ     '34 


94* 

*The  answer,  94,  was  recorded  during  Pause  16  at  the  point  indicated. 

First  reading  of  Problem  i  by  Subject  W  and  multiplication  direct  from  the 
problem  card  with  one  numeral  used  as  the  "base  of  operations." 

Immediately  after  the  first  reading,  the  subject  quickly  re-read  1000. 
After  Pause  22,  the  record  was  unsatisfactory. 

Plate  III  shows  the  first  reading  of  Problem  i ,  the  re-reading  of  both 
of  the  numerals,  and  the  process  which  was  followed  in  solving  the 
problem.  The  first  pause  fell  much  too  far  to  the  right  of  the  beginning 
of  the  line  and  two  regressive  movements  were  necessary  as  shown  by 
pauses  2  and  3.  The  numeral  47  was  evidently  carefully  read  in  the  two 
pauses  of  16/50  and  35/50  of  a  second  which  it  received.  The  first 
reading  of  the  problem  was  completed  with  Pause  9. 

The  re-reading  began  immediately  with  Pause  10  on  the  numeral 
47.  It  was  a  long  pause  of  greater  than  average  duration  notwith- 
standing the  fact  that  47  had  been  carefully  read  during  the  first  reading. 


44 


HOW  NUMERALS  ARE  READ 


Pause  1 1  probably  served  as  a  guiding  pause  in  the  long  move  from  47 
to  2.  This  last  numeral  was  re-read  during  Pause  12.  After  this  the 
subject  returned  to  the  numeral  47  and  made  it  the  base  of  the  operation 


A   wholesale  graii 


is    o 


• 

dng  of 


at  the  beginning  of  the  day 

767 


l».»4  (5  ijb       17 

bushels   of   whea 

9  *        '5 


,17  I?  I 

.     During   the  da 


ZO 


's  trafling   2J,765 
r  14 


38.31 


\(o 


1  1  Y  If 

T'Tf 

9          r        Tp|ia     to  '10 


1015 


The  numerals  were  copied  at  the  place  indicated  by  the  horizontal  line. 

First  reading  of  Problem  3  by  Subject  Hb  and  re-reading  the  numerals  for  copying. 
x  indicates  that  it  was  impossible  to  determine  with  precision  the  duration  of  the 
pauses. 

of  multiplication,  which  took  place  during  pauses  13  and  14.  When  the 
computation  was  completed,  the  eye  returned  to  the  vicinity  of  the 
numeral  2,  where  the  answer  was  recorded  during  Pause  16. 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES 


45 


Plate  IV  illustrates  the  first  reading  of  Problem  3  and  the  re-reading 
of  both  of  its  numerals  for  copying  by  Subject  Hb.  The  numeral 
243,987  was  given  a  detailed  whole  first  reading,  while  the  numeral 
21,765  in  the  same  line  was  passed  by  with  a  rapid  partial  first  reading. 
Of  the  large  number  of  pauses  used  in  reading  line  2,  six  were  placed  on 
the  digits  of  one  numeral.  The  first  reading  was  concluded  with  a 
relatively  rapid  reading  of  the  last  line. 

Immediately  after  the  first  reading,  pauses  28  and  29  were  used 
apparently  in  locating  the  first  numeral  and  Pause  30  in  locating  the 


PLATE  V 
5      < 


A  timber 


owns  on 


pllt  bf  p7  apres,  ajid  another  of 

19      '2      IT  16  >+        JO 


How 


much  groi  ad 


18 


he  owij  altogether  ? 


as. 


ia 


First  reading  of  Problem  2  by  Subject  Hb,  re-reading  the  first  numeral  and 
subsequent  re-reading  both  numerals  for  copying. 

place  where  it  was  to  be  copied.  During  pauses  31  and  32,  the  numeral 
was  re-read.  During  pauses  33  and  34  reference  was  again  made  to  the 
place  of  copying,  while  during  pauses  35  and  36  the  numeral  was  located 
again.  With  pauses  37  and  38  apparently  the  first  group  of  digits  was 
re-read,  and  the  second  group  of  digits  was  re-read  with  Pause  40.  The 
second  numeral  appears  to  have  been  located  with  pauses  42  and  43,  and 
it  was  then  re-read.  During  Pause  45  or  Pause  49  (or  during  both 
pauses)  21,765  was  copied.  After  Pause  49,  the  record  could  not  be 
followed  accurately. 

Plate  V  shows  that  Subject  Hb  gave  a  very  detailed  and  cautious 
first  reading  to  Problem  2.     Ten  pauses,  none  of  which  represented  a 


HOW  NUMERALS  ARE  READ 


regressive  movement,  were  required  to  read  the  first  line.  In  the  second 
line,  the  numeral  1643  was  given  a  whole  first  reading  with  pauses  n, 
12,  and  13. 

PLATE  VI 


1        1  f      J 

If  ond  telephone  company  uses 

G.5  <•        7  3 


II.  10 


thfe  yea 


•j 


,,  ,3 

,  and  another  company  in  the 


1  "I    'J  VI 

sahie  period  uses  617,4p3 

10  10  J         15    4 


20.  19      2 

4 


,  ht 


cross  bars       w  many  mor 


does  the  one  use  than  the  other? 


f      4«        33    37   3 


The  numerals  were  copied  at  the  place  indicated  by  the  horizontal  line. 

First  reading  of  Problem  5  by  Subject  M  and  re-reading  and  copying  the  two 
numerals,  x  indicates  that  it  was  impossible  to  determine  with  precision  the  duration 
of  the  pause. 

When  the  first  reading  was  finished  Hb  proceeded  immediately  to 
re-read  357,  apparently  moved  by  some  such  purpose  as  the  re-  location 
of  the  numeral  or  the  verification  of  one  of  its  digits.  He  then  re-read 
and  copied  1643  with  pauses  20  to  24,  inclusive,  whereupon  he  passed  to 
357,  which  he  re-read  with  Pause  26.  After  this  pause,  the  subject's 
attention  was  directed  to  the  margin  at  the  left  of  the  text  of  the  problem. 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES 


47 


During  the  first  reading  of  the  first  two  lines  of  Problem  5,  as  shown 
in  Plate  VI,  Subject  M  used  a  large  number  of  pauses  of  relatively  short 
durations.  The  last  line  of  the  problem  was  read  with  much  greater 
rapidity.  Each  of  the  long  numerals  was  given  a  partial  first  reading. 
M  approached  the  second  numeral  with  a  short  pause  and  left  it  with 
another  short  pause.  This  method  of  reading  long  numerals  was 
followed  by  him  in  the  case  of  isolated  numerals  in  several  instances  to 
which  attention  is  called  in  the  comment  on  plates  XX-XXI. 

Immediately  at  the  end  of  the  first  reading  he  re-read  the  numeral 
1,918,564  with  pauses  24  to  27,  inclusive,  and  copied  it  at  the  same  time 
at  the  point  indicated  without  moving  his  eyes  from  the  numeral.  The 
second  numeral  was  re-located  with  Pause  28,  and  the  first  numeral 
which  had  now  been  copied  was  located  with  Pause  29  in  order  to  deter- 
mine where  to  copy  the  second  numeral.  This  numeral  was  then  re-read 
and  copied  at  the  place  indicated  in  the  plate. 

Plate  VII  shows  Subject  Hb  reading  the  first  problem  very  cautiously. 
With  only  two  exceptions  every  word  and  numeral  in  the  problem  was 


a  dozfcn  what 


12. 


First  reading  of  Problem  i  by  Subject  Hb  and  re-reading  the  first  numeral  for 
copying. 

read  individually.  Such  a  large  number  of  pauses  is  in  sharp  contrast 
with  the  relatively  small  number  of  pauses  which  were  used  by  B  and  G 
in  reading  the  same  problem  as  shown  in  plates  VIII  and  X. 

After  the  first  reading,  which  was  finished  with  Pause  10,  the  numeral 
2  was  not  re-read.  The  first  numeral,  however,  was  re-read  with  Pause 
13  and  immediately  copied  on  the  problem  card  in  the  margin  to  the 
left  of  the  text. 

In  Plate  VIII  a  rapid  first  reading  of  the  text  of  the  problem  by 
Subject  B  is  observed.  Immediately  at  the  conclusion  of  the  first  reading, 


48  HOW  NUMERALS  ARE  READ 

the  subject  proceeded  to  the  first  numeral  which  was  used  as  the  "base 
of  operations"  for  the  multiplication  process  in  pauses  7  and  8.  The 
numeral  2  meanwhile  was  retained  in  memory.  The  answer  was  recorded 
during  Pause  9  immediately  below  the  word  "what"  in  the  text  of  the 
problem. 

PLATE  VIII 

/I 

dozen  what  will  2  dozeil  eggi  cost  ? 
is 


7      H  8< 

*The  answer,  94,  was  recorded  at  the  point  indicated  during  Pause  9. 

First  reading  of  Problem  i  by  Subject  B  and  the  process  of  computation  with 
the  first  numeral  as  the  "base  of  operations." 

In  Plate  IX  Subject  G  is  shown  reading  Problem  2  with  a  relatively 
small  number  of  pauses.  Only  four  pauses  were  used  in  the  last  line. 
The  first  numeral  was  read  partially  while  the  second  numeral,  1643,  was 
read  in  detail.  G  is  the  only  subject  who,  when  the  isolated  numerals 
were  being  read,  was  able  to  read  four  digits  in  detail  with  one  pause. 
The  instances  in  which  he  did  this  are  described  in  the  comment  concern- 
ing plates  XVI  and  XVII. 

When  the  first  reading  was  completed,  the  subject  turned  his  atten- 
tion to  the  numeral  357  and  used  it  as  the  "base  of  operations"  during 
the  process  of  addition.  This  operation  was  carried  on  "mentally" 
and  directly  from  the  problem  card.  One  or  two  digits  were  taken  at  a 
time,  the  computation  starting  from  the  right  with  Pause  12.  The 
answer  was  recorded  during  Pause  13  or  immediately  thereafter.  At 
the  end  of  Pause  13  the  subject  closed  his  eyes. 

The  plate  itself  supplies  ample  internal  evidence  of  the  fact  that 
the  numeral  1643  was  wholly  read  with  the  first  pause  which  was  18/50 
of  a  second  in  duration.  Such  is  undoubtedly  the  case  since  the  subject 
was  able  to  produce  the  correct  answer  without  ever  looking  at  the 
numeral  again. 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES  49 

The  reading  and  solving  of  Problem  i  by  Subject  G  is  described  in 
Plate  X.  The  conditions  of  the  problem  and  the  identity  of  the  numerals 
were  evidently  grasped  during  the  first  five  or  six  pauses.  The  numerals 
were  not  re-read  and  the  answer  was  recorded  during  either  Pause  8  or 
Pause  9,  or  during  both. 

During  pauses  6  and  8  the  subject  may  have  been  occupied  with 
' '  mental ' '  computation.  This  suggestion  is  offered  as  a  possible  explana- 
tion of  the  fact  that  Pause  6  was  not  located  on  any  reading  material, 
but  was  nevertheless  the  longest  of  all  pauses  used  in  connection  with  the 

PLATE  IX 

I.  f  ?  4 


timber  man  owns  onfe  plot  of  35 1 

fci  3  '2,  13 


A  tiipber  man  f>wns  on^  plot  of  2(57  acres,  and  ahothej  of 

II  7 


1643  acl-es. 


1643  acres.     How  mufh  ground  does  hf  own  altogether? 
a       uf 


First  reading  of  Problem  2  by  Subject  G  and  the  process  of  adding  the  two 
numerals. 

problem.  It  does  not  seem  probable  that  Pause  7  was  needed  by  this 
subject  as  a  re-reading  pause  in  this  problem.  If  computation  was 
proceeding  during  pauses  6  and  7,  evidently  the  eyes  were  roving  around 
without  direction.  Such  undirected  roving  occurred  very  rarely,  if  at 
all.  In  most  cases,  the  eyes  of  the  subjects  were  fixed  on  the  numerals 
which  were  involved  in  the  computation. 

The  first  reading,  the  re-reading,  and  the  solution  of  Problem  i  by 
Subject  M  are  shown  in  Plate  XI.  Apparently  the  subject  became 
confused  on  the  first  few  words  of  the  line  as  is  indicated  by  the  backward 
and  forward  movements  of  the  pauses.  Such  confusion  in  the  reading 
of  problems  was  found  in  but  very  few  instances. 


HOW  NUMERALS  ARE  READ 


After  the  first  reading,  some  of  the  last  words  of  the  problem  and 
the  numeral  2  were  re-read.  Such  re-reading  of  words  in  a  problem 
was  a  very  rare  occurrence  on  the  part  of  the  subjects  of  this  investigation. 
'The  answer  was  recorded  immediately  after  Pause  18  in  the  margin  of 
the  card  to  the  left  of  the  text  of  the  problem. 

PLATE  X 


1  *l  '[    "J 

At  47  cents  a  dozen  what  kvill  2 


dozen  eggi  cost  ? 

1C.' 


20 


J 


3  « 

te4* 

6  8 

*  The  answer,  94,  was  recorded  during  Pause  8  or  9. 

First  reading  of  Problem  i  by  Subject  G  and  the  process  of  computation 

PLATE  XI 

lHVlTl 

At  47  dents  a  dozen  what  will  JJ  dozefi  eggs  cosl|? 

15      9     S      «0     7   T       9 


14 


12. 


47 


14   II 


First  reading  of  Problem  i  by  Subject  M,  re-reading  words  and  the  process  of 
computation. 

During  the  first  reading  of  Problem  3,  as  shown  in  Plate  XII, 
Subject  G  gave  both  of  the  numerals  partial  first  readings.  Immediately 
at  the  end  of  the  first  reading,  which  was  finished  with  Pause  22,  he 
began  the  process  of  subtracting  the  second  numeral  from  the  first. 
With  Pause  23  he  located  the  first  numeral  and  with  Pause  24  perceived 
its  first  right-hand  digit.  He  then  quickly  glanced  at  the  first  right-hand 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES  51 

digit  of  the  second  numeral  with  Pause  25.  The  movements  back  and 
forth  between  the  two  numerals  continue  steadily,  one-digit  place  being 
computed  at  each  movement,  until  the  answer  was  recorded  immediately 
after  Pause  32. 


PLATE  XII 


A   whi 


(0 


[  1      I        1 

)lesble  graih  firm  haffl 

10  lj  »« 


at  the  beginning  of 

' 


1  1 

tihe  dakr 


Vl '         "L     1      "       1    1 

243,987    bushels   of   wheat.     Dbring  the  day's  trading  2U765 
zo    15     ii  io  t  >a  '<>  zd 


1  'I      "I      "I 

busUels  were  solti.     Ho\s| 


much  did 


hell 

13* 


they  then  have? 


14' 


Z5    jo 


First  reading  of  Problem  3  by  Subject  G  and  the  process  of  computation 

Attention  should  be  called  to  the  fact  that  since  all  of  the  digits  of 
the  answer  were  the  digit  2,  it  was  easier  for  the  subject  to  hold  the 
answer  in  memory  as  long  as  he  did  before  recording.  The  larger  numeral 
is  seen  to  have  given  one  more  pause,  not  counting  Pause  23,  and  the 
average  duration  of  its  pauses  was  greater  than  that  of  the  smaller 
numeral.  The  computation  began  and  ended  with  the  digits  of  the 
longer  numeral. 


HOW  NUMERALS  ARE  READ 


In  Plate  XIII  are  found  illustrations  of  pronounced  whole  first 
reading  of  numerals  by  Subject  H.  Even  the  text  of  the  problem  seems 
to  have  been  read  and  re-read  with  very  short  spans  of  attention  and 
with  meticulous  care. 

PLATE  XIII 


If  one  (telephone  dompani 


uses 


•6j  '5.  T  '*.  19       24          20 

the  yeJr,  and  ajiotheij  comparjy  in  the  Jam  ;  peri 


;  period  uses  pl7 

20  13       *Tt 


"  If       1 

one  ufee  flhan  the  oth^r  ? 

7  .I  J         Z9 


50,47 


43,,  40 


u'es 


Z.       2 


1301111     * 

*The  answer,  1301111,  was  recorded  at  the  point  indicated. 

First  reading  of  Problem  5  by  Subject  H  and  the  process  of  computation 

After  the  first  reading,  which  is  concluded  with  Pause  37,  the  compu- 
tation began  immediately  and  proceeded  in  a  manner  similar  to  that 
described  in  the  comment  concerning  Plate  XII.  The  numeral  1,918,564 
was  used  as  the  "base  of  operations";  the  computation  both  began 
and  ended  with  its  digits. 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES  53 

The  figures  of  the  answer,  1,301,111,  were  recorded  one  digit  at  a 
time,  as  they  were  produced  by  the  computation,  and  immediately  below 
the  words,  "use  than  the  other, "  in  the  text  of  the  problem.  Several  of 
the  pauses  were  used  in  directing  the  hand  as  it  recorded  the  digits  of 
the  answer.  This  was  true  of  pauses  on  each  of  the  two  numerals.  An 
effort  is  made  to  give  the  numbers  of  such  pauses  in  Table  XXVI. 

In  Plate  XIV  two  excellent  cases  of  pronounced  partial  first  readings 
are  found.  Although  the  numerals  are  five  and  six  digits  in  length, 
respectively,  nevertheless,  each  one  was  read  with  a  single  pause.  In 


PLATE  XIV 

3\   1    1          .'I.        1 

A  wholesale  grajn  firpi  hp,d  ap  the  beginning  of  tne  day 

10  IS 


\       ]    11   1 

alesale  grain  firm  had  at 

IT  14  I*          »3  9 


1       ']  "I        "I        ']     "I     '1 

243,987    bushels   of   whedt.     DurinJ;  the  dayfs  trading   21J65 

Z3  32  '4  's'  iJ  14  I* 


'1  "1  1  1          "1 

bushels  were  sold.     How  much  did  they  then  rmve? 

13  /o'  ii '  n'l  3 

First  reading  of  Problem  3  by  Subject  W  with  partial  reading  of  numerals 

this  plate  a  clear  illustration  of  rapid  reading  of  the  last  line  of  a  problem 
is  also  found.  Only  five  pauses  were  required  for  reading  line  3  and 
their  durations  were  less  than  this  subject's  average  pause-duration  on 
words  as  given  in  Table  XVIII. 

Plate  XV  exhibits  the  process  of  solving  Problem  5  as  it  was  carried 
on  by  Subject  G.  His  procedure  was  similar  to  that  of  Subject  H  which 
is  described  in  the  comment  accompanying  Plate  XIII. 

The  use  to  which  Subject  G  put  each  individual  pause  in  the  compu- 
tation is  described  in  Table  XXVI.  An  important  difference  should  be 
noted  between  the  procedures  of  subjects  G  and  H  in  solving  Problem  5. 


54 


HOW  NUMERALS  ARE  READ 


Subject  G,  as  is  shown  by  the  location  of  pauses  23  and  24  of  Plate  XV, 
began  the  computation  by  taking  the  first  right-hand  digit  of  617,453 
and  proceeded  to  relate  it  to  the  corresponding  digit  of  1,918,564. 
He  continued  the  process  by  moving  from  right  to  left  Subject  H,  on 

PLATE  XV 

*Ll         1   1  \  \    1     \    'J 

If  one  telephone  company  uses  1,D18,564  cros|  bars  jhmngj 

12     W  9  J          3  J  «'  J  T' 


,.  "1  I  "I 

the  year,  and  another  company  tn  the  same  peribd  uses  6 

J  13'  ?'  J 


1     1 

5^7,453 

iJ         3 


cross 


1  1  '1    1    T    1    T 

rsL  how  mankr  more  poes  the  bne  use  flhan  thfe  other  ? 
V     \i        to*          M'  s  ««  a 


1301111 

*The  answer,  1301111,  was  recorded  at  the  point  indicated. 

First  reading  of  Problem  5  by  Subject  G  and  the  process  of  computation 

the  other  hand,  began  the  computation  by  taking  the  first  right-hand 
digit  of  1,918,564,  and  proceeded  to  find  the  corresponding  digit  of 
617,453.  He  continued  the  process,  as  did  Subject  G,  by  moving  from 
right  to  left.  Both  subjects,  however,  appear  to  have  emphasized  the 
larger  numeral  as  the  "base  of  operations."  The  details  are  given  in 
Table  XXVI. 


DESCRIPTION  OF  THE  EYE-MOVEMENT  STUDIES  55 

This  concludes  the  general  description  of  the  photographic  records. 
In  the  following  divisions  of  the  report  various  phases  of  the  reading  of 
numerals  will  be  discussed  in  greater  detail.  In  the  next  chapter  a 
description  is  given  of  the  first  reading  of  the  problems  material.  Chapter 
viii  provides  a  discussion  of  the  re-reading  of  problems  and  the  processes 
of  computation.  The  reading  of  isolated  numerals  is  described  in  chapter 
ix.  In  the  last  chapter  the  performance  of  the  subjects  of  this  investiga- 
tion is  compared  with  that  of  the  subjects  of  an  important  investigation 
by  another  author,  and  finally  the  report  concludes  with  a  discussion  of 
the  differences  in  the  demands  which  are  made  upon  the  attention  of 
readers  by  the  three  different  types  of  reading-materials. 


CHAPTER  VII 

FIRST  READING  OF  NUMERALS  IN  PROBLEMS 
I.      INTRODUCTION 

When  examining  plates  I— XV,  which  reproduce  the  lines  of  the  prob- 
lems as  they  were  read  and  which  locate  within  the  lines  the  pauses 
as  they  occurred  in  the  readings,  the  unusually  large  number  of  pauses 
per  line  stands  out  very  conspicuously.  The  average  number  of  pauses 
per  line  for  all  subjects  is  8.08;  and  there  are  individual  lines  in  which 
as  many  as  10,  n,  12,  13,  and  even  14  pauses  are  found.  The  large 
number  of  pauses  appears  all  the  more  remarkable  when  it  is  remembered 
that  all  of  the  readers  were  advanced  graduate  students,  who  are  entirely 
familiar  with  simple  arithmetical  problems,  and  who  would  be  expected 
to  qualify  as  better  than  average  readers. 

Attention  should  be  called  at  this  point  to  the  fact,  which  is  given 
more  detailed  treatment  in  a  later  section,  that  the  subjects  of  this  study 
were  not  slow  readers.  It  appears  that  there  is  good  ground  for  assuming 
that  the  reading  of  arithmetical  problems  is  more  difficult  than  the 
reading  of  ordinary  prose.  The  question  suggests  itself,  therefore:  Did 
the  two  elements  of  which  the  problems  are  composed,  namely,  the 
numerals  and  the  accompanying  words,  make  equal  demands  upon  the 
attention  of  the  individuals  who  read  them  in  this  study  ?  The  data, 
by  means  of  which  comparisons  may  be  drawn  between  the  numerals 
and  the  words,  with  respect  to  average  duration  of  pauses,  average 
number  of  letters  or  digits  included  per  pause,  and  the  percentage 
which  the  regressions  are  of  the  total  number  of  pauses,  are  presented 
in  tables  XVII-XIX. 

2.      COMPARISON    BETWEEN    THE    READING    OF    NUMERALS    AND 
WORDS    IN   PROBLEMS 

It  is  evident  from  a  glance  at  Table  XVII  that  there  is  a  very  great 
difference  in  the  average  ranges  of  acts  of  perception  according  as  digits 
in  numerals  or  letters  in  words  are  read.  In  the  readings  of  all  of  the 
subjects  the  average  number  of  digits  included  by  a  pause  on  numerals 
was  less  than  the  average  number  of  letters  included  by  a  pause  on  words. 
The  disparity  between  these  averages  is  slightly  greater  in  the  cases  of 
the  three  whole  first  readers  B,  H,  and  Hb,  all  of  whom  show  shorter 

56 


FIRST  READING  OF  NUMERALS  IN  PROBLEMS 


57 


ranges  of  perception  of  digits  than  the  three  other  subjects,  who  are  par- 
tial first  readers.  Even  the  partial  first  readers,  however,  in  every  case, 
perceived  on  the  average  less  than  half  as  many  digits  as  letters  per 
pause. 

The  explanation  of  this  shorter  range  of  perception  for  numerals 
than  for  words,  when  both  occur  in  the  same  arithmetical  problem  is 
probably  the  same  as  that  given  by  Dearborn  in  accounting  for  the  short 
"number  span  of  attention,"  which  he  had  noted.1  The  digits  in 
numerals  do  not  appear  constantly  in  the  same  combinations  as  do  the 
letters  in  words.  In  consequence,  the  numerals  in  their  continually  new 
combinations  of  digits  make  larger  demands  upon  the  attention  of 
readers.  Every  individual  digit  is  significant  in  itself  and  must  be  noted ; 
and  all  of  the  digits  must  be  viewed  in  combination  before  the  numeral 

TABLE  XVII 

AVERAGE  NUMBER  OF  DIGITS  INCLUDED  IN  A  PAUSE  ON  NUMERALS  CONTRASTED  WITH 

AVERAGE  NUMBER  OF  LETTERS  INCLUDED  IN  A  PAUSE  ON  WORDS  DURING 

FIRST  READING 


SUBJ 

ECTS 

AVERAGE 

FOR 

G 

M 

w 

B 

H 

Hb 

ALL 
SUBJECTS 

Average  number  of  digits  included  in  a  pause  on 
numerals 

i  81 

i  88 

i  88 

2    *8 

Average  number  of  letters  included  in  a  pause  on 
words 

7  30 

e    xg 

6   4.7 

NOTE. — Each  subject  read  the  five  problems  which  included  12  numerals  totaling  47  digits,  and 
in  words  totaling  482  letters. 

is  completely  read.  Words,  however,  as  several  investigations  of  the 
span  of  perception  have  shown,  are  perceived  as  wholes.  The  letters 
appear  and  reappear  in  the  same  regular  combinations,  which  become 
familiar  in  the  earlier  years  of  schooling.  Readers  have  become  accus- 
tomed to  them  as  words  and  are  able  to  proceed  easily  with  whole  words 
as  units  of  perception. 

As  noted  in  the  foregoing  paragraph,  the  pauses  on  numerals  are 
more  concerned  with  analysis  and  combination  of  the  component  digits 
than  the  pauses  on  words  are  with  similar  processes  with  the  letters. 
Such  a  difference  would  be  expected  to  make  itself  evident  in  a  greater 
average  duration  for  the  pauses  on  numerals  than  for  the  pauses  on 
words.  The  data  which  are  displayed  in  Table  XVIII  justify  such  an 

*W.  F.  Dearborn,  "The  Psychology  of  Reading,  An  Experimental  Study  of 
the  Reading  Pauses  and  Movements  of  the  Eye,"  Columbia  University  Contributions 
to  Philosophy  and  Psychology,  Vol.  XIV,  No.  i  (1906),  pp.  70-71.  New  York:  The 
Science  Press. 


HOW  NUMERALS  ARE  READ 


expectation.  With  each  of  the  several  subjects,  it  is  seen  that  the  average 
duration  of  the  pauses  on  numerals  was  decidedly  greater  than  the 
average  duration  of  the  pauses  on  words.  The  average  for  all  subjects 
of  the  average  pause-durations,  when  numerals  were  read,  is  approxi- 
mately 40  per  cent  greater  than  the  same  average  duration  when  words 
were  being  read. 

TABLE  XVIII 

AVERAGE  DURATION  OF  PAUSES  IN  FIFTIETHS  OF  A  SECOND  ON  NUMERALS  CONTRASTED 
WITH  AVERAGE  DURATION  OF  PAUSES  ON  WORDS  DURING  FIRST  READING 


SUBJECTS 

AVERAGE 

FOR 

ALL 
SUBJECTS 

G 

M 

W 

B 

H 

Hb 

Total  number  of  pauses     ^Numerals.  . 
used  by  subject  in  reading/  Words  .... 
i.  Average  duration  of  pauses  on  nu- 
merals     

14 
66 

13-92 
3-92 

10.72 
2.27 

20 
81 

15.20 

4-54 

9-87 
1.41 

16 
7i 

15-31 
5-35 

13.02 
3.38 

26 
61 

18.46 
5-92 

9.18 
1.16 

27 
93 

13.30 
3-97 

11.74 
4-13 

11 

13.48 
5-0 

10.99 
2-94 



14.98 
4-77 

10.92 
2-55 

Average  variation 

2.  Average  duration  of  pauses  on  words 
Average  variation  

A  comparison  between  the  numerals  and  the  words  in  respect  to 
the  percentage  which  the  number  of  regressive  pauses  is  of  the  total 
number  of  pauses  for  a  subject,  yields  further  evidence  of  the  greater 
reading-demands  made  by  numerals.  In  the  cases  of  subjects  M,  B,  H, 
and  Hb,  as  found  in  Table  XIX,  decidedly  larger  percentages  of  regres- 

TABLE  XIX 

PERCENTAGE  OF  REGRESSIVE  PAUSES  ON  NUMERALS  CONTRASTED  WITH  PERCENTAGE 
OF  REGRESSIVE  PAUSES  ON  WORDS  DURING  FIRST  READING 


G 

M 

W 

B 

H 

Hb 

Total   number   of   regressive    pauses  /Numerals  
located  by  subject  on  \Words  
Percentage  which  the  number  of  regressive  pauses  on: 
i.  Numerals  is  of  the  total  number  of  pauses  on 

2 
II 

14.28 

6 

10 

30.0 

I 
4 

6.25 

7 
8 

26.9 

10 

22 
4O.O 

5 

i 

20.  o 

2.  Words  is  of  the  total  number  of  pauses  on  words 

16.6 

12.3 

5.63 

13-1 

23.65 

1.19 

SUBJECTS 


sive  pauses  appear  in  the  case  of  the  numerals  than  upon  the  accompany- 
ing words.  The  explanation  of  such  differences  probably  lies  in  the 
difficulty  of  reading  in  the  same  lines,  materials  which  call  for  such  differ- 
ent ranges  of  attention  and  durations  of  pauses  as  did  the  numerals 
and  the  words  in  these  problems.  When  proceeding  at  the  rate  of 
reading  and  with  the  range  of  perception  which  is  adapted  to  words, 


FIRST  READING  OF  NUMERALS  IN  PROBLEMS  59 

the  subject  apparently  passes  over  some  of  the  numerals  with  a  reading 
which  does  not  satisfy  him,  and  he  immediately  returns  to  read  or  to 
re-read  all  or  a  part  of  the  numeral. 

3.      PARTIAL   AND   WHOLE   FIRST   READING   OF   NUMERALS 

Whole  first  reading  was  defined  in  the  first  preliminary  study  of 
the  investigation  to  include  such  readings  of  numerals  during  the  first 
reading  of  a  problem  as  noted  the  character  of  the  numeral  and  the 
identity  and  place  in  the  numeral  of  each  individual  digit.  Any  reading 
of  a  numeral  which  did  not  include  these  items  was  called  a  partial  first 
reading.  In  Table  XX  the  kind  of  reading  given  each  of  the  twelve 
numerals  in  the  problems  by  each  of  the  several  subjects  is  described 
in  detail.  The  data  which  are  included  in  this  table  are  based  upon 
introspective  observations  concerning  their  readings  by  several  of  the 
subjects,  and  upon  inferences  which  were  drawn  directly  from  the 
plates.  With  the  longer  numerals  a  very  small  number  of  pauses  of 
short  duration  in  some  instances  gave  indisputable  evidence  of  partial 
reading.  In  several  of  the  records  answers  to  problems  which  included 
shorter  numerals  were  computed  and  recorded  when  the  numerals  had 
been  read  only  on  the  first  reading;  obviously  such  readings  were  whole 
first  readings.  A  few  readings  could  not  be  placed  with  certainty  in 
either  category  and  are,  therefore,  marked  D,  which  means  doubtful. 
Plates  II,  IV,  VI,  XII,  and  XIV  present  instances  in  which  numerals 
received  partial  first  readings,  and  plates  I,  IV,  IX,  and  XIII  show 
other  instances  in  which  numerals  received  whole  readings. 

There  are  marked  differences  between  partial  and  whole  first  readings 
of  the  longer  numerals  in  respect  to  the  number  of  pauses  per  numeral 
and  the  total  time  required  for  reading  the  numeral.  These  differences 
are  quickly  apparent  when  Table  XX  and  the  plates  which  bear  illustra- 
tions of  the  two  methods  are  studied  in  detail.  Illustrations  of  both 
methods  of  reading  are  found  in  Plate  IV  which  represents  the  reading  of 
Problem  3  by  Subject  Hb.  In  this  problem  one  of  the  numerals,  243,987, 
was  given  a  whole  first  reading  which  included  six  pauses  and  measured 
a  total  reading-time  of  101/50  seconds,  while  the  other  numeral,  21,765, 
was  given  a  partial  reading  which  included  only  one  pause  with  the 
duration  of  14/50  of  a  second. 

Further  emphasis  is  given  to  the  difference  between  partial  and  whole 
reading  of  numerals  by  a  comparison  of  the  first  readings  of  the  numerals 
in  the  problems  with  the  readings  of  the  numerals  isolated  in  lines.  As 
prescribed  by  the  conditions,  the  readings  of  the  isolated  numerals 


60 


HOW  NUMERALS  ARE  READ 


II 

i 


| 

I    N 

MM                           CO           M            CO 

2 

a 
c 

I 

3 

^                ~-i    M 

0 

PH 

s     Si-- 

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FIRST  READING  OF  NUMERALS  IN  PROBLEMS  6 1 

were  of  the  quality  of  whole  readings.  The  data  are  arranged  in  Table 
XXI  for  such  a  comparison  between  the  two  sets  of  numerals  in  each 
of  the  several  digit-lengths  with  respect  to  the  average  number  of  pauses 
per  numeral,  the  average  pause-duration,  and  the  average  time  required 
for  reading  individual  numerals.  In  respect  to  each  of  the  three  points 
named  for  comparison  the  isolated  numerals  are  found  to  have  larger 
averages  than  the  problem  numerals  in  each  of  the  several  digit-lengths. 
If  only  those  problem  numerals  which  were  given  partial  first  readings 
were  included  in  the  comparison,  the  differences  between  the  two  sets 
of  numerals  would  be  appreciably  greater  than  they  are. 

On  the  other  hand  a  certain  degree  of  qualification  is  attached 
necessarily  to  the  significance  of  the  large  differences  found  because  of 
the  effect  probably  produced  on  the  reading  of  the  isolated  numerals 
by  one  of  the  conditions  under  which  they  were  read.  The  slight 
articulation  used  in  reading  the  isolated  numerals  presumably  acted 
to  diminish  the  speed  with  which  they  were  read.  It  should  be  noted 
also  that  all  of  the  subjects  gave  evidence  of  greater  interest  in  reading 
and  solving  the  problems  than  in  reading  the  numerals  isolated  in  lines. 
The  probable  effect  of  such  great  interest  in  the  problems  was  to  stimulate 
the  readers  to  a  more  rapid  rate  of  reading  the  numerals  in  problems 
than  the  numerals  of  corresponding  digit-length  which  were  isolated 
in  lines. 

Proper  allowances  should  be  made  for  the  differences  which  were 
produced  in  the  readings  of  the  two  sets  of  numerals  by  such  variations 
in  conditions  as  are  named  above.  When  this  allowance  is  made,  it  is 
found  that  the  whole  first  readings  which  were  given  the  longer  numerals 
in  the  problems  by  the  whole  first  readers  include  numbers  of  pauses  and 
total  reading-times  per  numeral  which  are  similar  to  those  found  in  the 
readings  of  the  isolated  numerals.  The  whole  first  reading  of  numerals 
in  problems  is  evidently  similar  in  kind  to  the  reading  of  numerals 
isolated  in  lines. 

As  was  found  in  the  first  preliminary  study,  marked  differences 
appear  between  the  shorter  and  longer  numerals  in  respect  to  the  number 
of  times  in  which  they  were  partially  and  wholly  read  during  first  read- 
ings. The  one-  and  two-digit  numerals  were  read  in  detail  in  most 
instances  in  the  present  study  as  they  were  in  previous  studies.  Only 
one  pause,  for  the  most  part,  was  required  for  the  reading  of  the  shorter 
numerals  and  the  durations  of  such  pauses  ran  as  low  as  5/50  and  8/50 
of  a  second.  The  longer  numerals  on  the  other  hand  received  a  slightly 
greater  number  of  partial  readings  than  whole  readings.  The  partial 


62 


HOW  NUMERALS  ARE  READ 


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FIRST  READING  OF  NUMERALS  IN  PROBLEMS  63 

readings  for  the  five-digit  numerals  included  one  pause  for  the  most 
part;  and  in  the  six-  and  seven-digit  numerals  two  pauses  were  included 
in  most  instances.  In  respect  to  proportion  of  partial  readings  received, 
the  compact  group  of  three  numerals  in  Problem  4,  which  are  placed 
one  immediately  after  the  other,  may  be  classified  with  the  longer 
numerals. 

The  familiar  numeral  1000  has  received  a  kind  of  reading  which 
distinguishes  it  from  other  numerals  of  the  same  length,  wherever  it 
has  appeared  in  the  preliminary  studies.  In  the  present  study  1000 
was  given  whole  first  readings  without  exception.  In  no  instance  was 
more  than  one  pause  required  for  the  reading,  and  the  lengths  of  the 
pauses  closely  approximate  average  pause-durations.  For  whole  first 
readings  of  other  numerals  of  four-digit  length,  which  are  found  in  the 
problems,  two  or  three  pauses  were  required.  Unlike  the  other  four- 
digit  numerals,  1000  is  evidently  read  as  a  whole,  as  words  are  read. 

It  is  probable  that  the  form  of  1000,  which  is  that  of  the  digit  "  i," 
followed  by  the  very  obvious  group  of  three  "o"  digits  is  easier  of 
perception  than  any  ordinary  group  of  four  digits.  It  is  probably 
easier  of  perception  than  any  numeral  which  is  made  up  of  a  group  of 
four  digits  all  of  which  are  the  same  digit.  The  frequency  of  appearance 
of  1000  in  the  experience  of  the  average  reader  is  probably  greater  than 
that  of  any  other  single  combination  of  four  digits.  By  virtue,  therefore, 
of  the  obviousness  of  its  structure  and  by  virtue  of  the  frequency  of  its 
use  the  numeral  1000  has  become  familiar  to  the  average  reader  in  the 
same  sense  that  words  are  familiar,  and  in  consequence  is  read  as  words 
are  read. 

The  number  of  digits  which  it  is  possible  to  read  partially  in  one 
pause  of  partial  reading  is  relatively  large.  The  five  digits  of  the  five- 
digit  numerals  were  read  with  one  pause  in  three  of  the  four  instances 
when  they  were  partially  read.  One  of  the  six-digit  numerals  was  read 
one  time  with  one  pause,  as  is  shown  in  Table  XX.  The  six  and  seven 
digits  of  the  six-  and  seven-digit  numerals  were,  however,  read  with  two 
pauses  in  most  instances  when  they  were  partially  read.  The  usual 
number  of  digits  read  at  one  pause  when  such  longer  numerals  were 
read  partially  is,  therefore,  either  three  or  four. 

In  the  preliminary  study  which  was  concerned  with  the  range  of 
recall  from  the  first  reading  of  numerals  in  problems  it  was  shown  that 
in  a  great  preponderance  of  instances  the  subjects  were  able  to  recall 
at  least  the  number  of  digits  in  a  numeral.  In  view  of  this  fact  it  seems 
reasonable  to  assume  that  one  or  more  of  the  subjects  who  read  the 


64 


HOW  NUMERALS  ARE  READ 


five-digit  numerals  partially  and  with  a  single  pause,  were  able  to  read 
all  five  of  the  five  digits  at  one  pause,  at  least  to  the  extent  of  noting 
that  a  numeral  was  there  and  that  its  length  was  five  digits.  Apparently, 
therefore,  it  is  possible  for  subjects  to  read  to  this  extent  as  many  as 
five  digits  at  one  act  of  perception. 

4.      THE    SEVERAL    SUBJECTS    AS    PARTIAL    AND    WHOLE     FIRST    READERS 

The  largest  factor  in  determining  whether  the  longer  numerals  shall 
be  given  partial  or  whole  first  readings  is  found  in  the  attitude  of  indi- 
vidual subjects  toward  these  numerals.  The  same  numerals  were  given 
partial  readings  by  some  of  the  subjects  and  whole  readings  by  others. 
An  examination  of  the  data  in  Table  XX  makes  it  evident  that  some  of 
the  subjects  gave  partial  readings  with  noticeable  consistency  to  the 
longer  numerals,  while  other  subjects  with  approximately  equal  con- 
sistency gave  whole  readings  to  these  numerals.  Subjects  G,  M,  and 
W  clearly  exhibit  the  former  tendency  and  may,  therefore,  be  classified 
as  partial  first  readers,  while  B,  H,  and  Hb  illustrate  the  latter  tendency 
and  may  be  called  whole  first  readers. 

5.      RELATIVE  VALUE  OF  PARTIAL  AND  OF  WHOLE  FIRST  READING 

Any  determination  of  the  relative  value  of  whole  and  partial  reading 
as  methods  of  reading  numerals  during  the  first  reading  of  a  problem 
will  be  concerned  to  a  large  extent  with  the  question  as  to  which  of  the 
methods  is  more  economical  of  the  reader's  time.  This  question  resolves 
itself  in  large  part  into  a  comparison  between  the  partial  and  whole  first 
readers  in  respect  to  the  total  time  required  to  read  all  of  the  numerals 
of  the  problems.  The  three  partial  readers  G,  M,  and  W,  as  is  readily 
seen  by  inspection  of  Table  XXII,  used  decidedly  shorter  total  times 

TABLE  XXII 

READING  OF  NUMERALS  IN  PROBLEMS  BY  PARTIAL  FIRST  READERS  CONTRASTED  WITH 

READING  OF  NUMERALS  IN  PROBLEMS  BY  WHOLE  FIRST  READERS 

(Time  unit  =  1/50  of  a  second) 


SUBJECTS 

Partial  First  Readers 

Whole  First  Readers 

G 

M 

W 

B 

H 

Hb 

Total  time  required  to  read  all  numerals  
Total  number  of  pauses  required  to  read  all  numerals 

195 

14 
13.92 

304 

20 
15.20 

245 
16 
15-31 

480 
26 
18.46 

359 
27 
13.30 

337 

25 

13.48 

NOTE. — Each  subject  read  all  five  problems  which  included  twelve  numerals. 


FIRST  READING  OF  NUMERALS  IN  PROBLEMS 


for  the  first  reading  of  all  of  the  numerals  than  the  whole  readers  B,  H, 
and  Hb.  Partial  first  reading,  therefore,  in  so  far  as  time  required  for 
reading  the  numerals  is  concerned,  was  undoubtedly  the  more  economical 
of  the  two  methods. 

The  words  of  the  problems  were  also  read  more  rapidly  by  the 
partial  first  readers,  although  the  case  is  not  as  clear  for  the  words  as 
it  is  for  the  numerals.  The  details  are  given  in  Table  XXIII.  The 
fastest  reader  of  the  words  is  the  whole  first  reader,  Subject  B,  who  is 
also  the  slowest  reader  of  the  numerals.  His  case  appears  to  be  very 
exceptional  and  no  adequate  explanation  is  found  in  the  records,  nor 
was  the  subject  himself  able  to  account  for  his  relatively  high  speed 
with  words.  After  B,  the  two  partial  readers,  G  and  M,  have  the 
fastest  records.  The  slowest  record  was  made  by  the  whole  first  reader 
H.  By  virtue  of  relatively  higher  speeds  with  both  numerals  and  words 
the  partial  first  readers  completed  the  first  reading  of  the  problems  in 
shorter  total  reading-times  than  the  whole  readers. 

TABLE  XXIII 

READING  OF  WORDS  IN  PROBLEMS  BY  PARTIAL  FIRST  READERS  CONTRASTED  WITH 

READING  OF  WORDS  IN  PROBLEMS  BY  WHOLE  FIRST  READERS 

(Time  unit  =  1/50  of  a  second) 


SUBJECTS 

Partial  First  Readers 

Whole  First  Readers 

G 

M 

W 

B 

H 

Hb 

Total  time  required  to  read  all  words  of  problems  .  .  . 
Total  number  of  pauses  required  to  read  all  words  of 
problems                  

708 

66 
10.72 

800 

81 
9.87 

925 
7i 

13-02 

560 

61 
9.18 

1092 

93 
11.74 

923 

84 
10.99 

Average  pause-duration                               .           ... 

NOTE. — Each  subject  read  all  five  problems  which  included  in  words. 

Another  basis  may  be  found  upon  which  significant  inferences  can 
be  drawn  as  to  the  relative  value  of  the  two  methods  of  first  reading  of 
numerals.  It  is  possible  to  draw  a  comparison  between  the  several 
individuals,  who  use  the  one  or  the  other  of  the  two  methods,  in  respect 
to  their  rates  of  reading  with  reading  materials  other  than  arithmetical 
problems.  Stated  in  other  terms  the  point  in  question  is:  Which  of 
the  two  methods  was  used  by  the  readers  who  exhibit  higher  rates  of 
reading  in  other  materials.  The  data  concerning  the  rates  of  speed 
with  which  the  subjects  read  the  ordinary  prose  selection  may  be  used  in 
this  comparison.  With  these  materials  the  average  reading-time  per 


66  HOW  NUMERALS  ARE  READ 

line  for  subjects  M  and  G  were  44.88/50  and  52.52/50  seconds,  respec- 
tively, and  for  subjects  W,  H,  and  B  they  were  75.21/50,  75/50,  and 
80.78/50  of  a  second,  respectively.  It  is  clear,  therefore,  that  the  faster 
readers  of  the  ordinary  prose  selection  used  the  partial  method  of  reading 
numerals  on  the  first  reading,  and  two  of  the  three  slower  readers  of  the 
ordinary  prose  used  the  whole  method  of  reading  numerals. 

The  more  rapid  reading  of  the  partial  first  readers  is  due  to  the  fact 
that  for  the  most  part  they  used  fewer  pauses  in  reading  the  same 
materials  than  the  whole  readers.  Such  is  the  explanation  of  the  greater 
speeds  whether  the  materials  which  were  used  were  the  numerals  or 
words  of  a  problem,  or  the  lines  of  the  ordinary  prose  selection.  A 
smaller  number  of  pauses  also  explains  the  exceptionally  short  total 
reading-time  of  Subject  B  on  the  words  of  the  problems. 

6.      DEVELOPMENT    OF   THE    METHOD    OF   PARTIAL  FIRST   READING 

The  large  differences  between  the  methods  of  partial  and  whole 
first  reading  in  the  several  important  respects  to  which  attention  has 
been  directed  in  this  immediate  study  and  in  previous  studies  of  the 
investigation  can  be  satisfactorily  explained  only  in  the  light  of  large 
differences  in  attitude  on  the  part  of  the  individuals  who  use  respectively 
the  one  or  the  other  of  the  two  methods.  The  evidence  is  strong  that 
partial  and  whole  first  readers  entertain  very  different  attitudes  toward 
the  numerals  when  the  problem  is  being  read  for  the  first  time.  The 
fact  that  such  differences  between  the  two  groups  do  exist  does  not 
necessarily  imply  that  the  members  of  either  group  are  conscious,  either 
of  their  own  peculiar  attitude,  or  of  the  existence  of  a  different  attitude. 
Most  of  the  subjects  of  the  investigation  were  not  conscious  of  their 
attitudes.  The  implication  is,  rather,  that  through  long  experience  with 
problems  these  subjects  have  developed  in  an  empirical  way  two  widely 
differing  sets  of  habits  of  reading  numerals  during  the  first  reading  of  a 
problem. 

When  first  learning  to  read  numerals  in  problems,  these  subjects, 
as  most  individuals  probably  do,  proceeded  to  read  the  numerals  very 
slowly,  as  they  came  to  them,  and  one  digit  at  a  time.  The  whole 
reading  attitude  toward  numerals  would,  therefore,  be  the  attitude 
more  natural  for  beginners  in  reading  arithmetical  problems.  In  the 
course  of  extensive  experience,  apparently,  some  of  the  individuals  were 
more  impressed  than  others,  with  the  differences  between  words  and 
numerals  in  respect  to  the  rates  of  speed  which  were  found  practicable 
in  reading  them.  These  individuals  were  thus  stimulated  to  learn  a  new 


FIRST  READINGS  OF  NUMERALS  IN  PROBLEMS  67 

method  of  procedure  with  the  numerals,  which  would  make  for  a  quicker 
disposition  of  them  in  reading  problems.  Such  a  new  procedure  could 
not  consist  simply  of  a  radical  increase  of  the  span  of  perception  to  include 
larger  units  of  recognition,  as  had  been  done  with  word  materials  when 
whole  words  and  phrases  came  to  be  taken  as  units  of  perception. 
Perception  of  the  larger  numerals  by  numeral  wholes  appears  to  be 
quite  impracticable  because  of  the  nature  of  numerals  as  continually 
varying  combinations  of  digits. 

The  plan  that  was  learned  consists  of  a  rapid  passage  over  the 
numeral  during  which  time  details  are  skipped  and  only  the  most 
outstanding  facts  concerning  it  are  gathered.  Such  is  the  procedure 
which  is  designated  as  partial  first  reading  and  in  preliminary  sections 
of  the  investigation  the  identity  of  the  first  digit  of  a  numeral  and 
recognition  of  the  number  of  its  digits  were  found  to  be  facts  of  the 
outstanding  nature  referred  to.  Various  ranges  of  partial  first  reading 
came  to  find  empirical  acceptance  in  the  course  of  the  development  of 
the  habit  of  partial  reading  by  the  natural  trial-and-error  method  of 
learning.  Such  changes  in  procedure  were  probably  able  to  be 
accomplished  with  little  or  no  embarrassment  to  individuals  in  the 
practical  solving  of  problems  because  of  the  almost  invariable  habit  of 
re-reading  the  numerals  after  the  first  reading  of  a  problem. 

7.      SUMMARY   OF   CONCLUSIONS 

The  chapter  may  be  summarized  as  follows: 

1.  The  numerals   of  problems  make  greater  demands   upon   the 
attention  of  readers  than  do  the  accompanying  words,  as  is  shown  by 
the  following  facts :  (a)  The  average  number  of  digits  included  by  a  pause 
on  numerals  is  decidedly  smaller  than  the  average  number  of  letters 
included  by  a  pause  on  words,     (b)  The  average  duration  of  pauses  on 
numerals  is  greater  than  the  average  duration  of  pauses  on  words  in  the 
cases  of  all  subjects,     (c)  The  percentage  of  regressive  pauses  on  numerals 
is  greater  than  the  percentage  of  such  pauses  on  words.     The  explanation 
of  the  greater  demands  of  the  numerals  probably  lies  in  the  fact  that  the 
combinations  of  digits  in  numerals  are  continually  different,  whereas 
combinations  of  letters  in  words  remain  stable. 

2.  More  pauses  per  numeral  and  greater  total  reading- times  per 
numeral  are  required  for  whole  first  reading  of  numerals  than  for  partial 
first  reading. 

3.  Shorter  numerals  are  given  whole  first  readings  almost  invariably. 
Longer  numerals,  on  the  other  hand,  are  given  whole  or  partial  readings 
according  as  the  subjects  who  read  them  are  whole  or  partial  readers. 


68  HOW  NUMERALS  ARE  READ 

4.  The  numeral  1000  is  regularly  read  as  if  it  were  a  word  rather  than 
a  numeral. 

5.  The  subjects  divide  themselves  into  groups  of  partial  first  readers 
and  of  whole  first  readers  according  as  they  read  the  longer  numerals 
by  the  partial  or  whole  method. 

6.  The  partial  method  is  more  economical  than  the  whole  method 
in  point  of  total  time  required  to  read  the  numerals  during  the  first 
reading  of  a  problem. 

7.  The  subjects  who  read  the  ordinary  prose  selection  more  rapidly 
use  the  partial  method  of  reading  the  longer  numerals  for  the  most  part. 

8.  The  more  rapid  rates  of  reading  in  both  the  words  and  numerals 
of  problems  and  in  the  ordinary  prose  selection  are  exhibited  by  subjects 
who  use  the  smallest  number  of  pauses  in  such  readings. 

9.  Partial  first  reading  of  numerals  is  probably  learned  empirically 
as  a  method  of  more  rapidly  disposing  of  the  numerals  in  reading.     The 
essential  characteristics  of  the  method  are   skipping  the  details  of  a 
numeral  and  recognizing  only  the  most  outstanding  facts  concerning  it. 


CHAPTER  VIII 

RE-READING  AND  COMPUTATION 
I.      TWO   TYPES   OF   RE-READING   OF   NUMERALS 

Re-reading  from  a  problem  takes  place  immediately  upon  completion 
of  the  first  reading.  Two  distinct  types  of  re-reading  of  numerals  appear. 
The  two  types  are  distinguished  by  differences  in  function.  In  some 
cases  such  differences  are  disclosed  directly  by  reports  which  were  made 
by  the  subjects  themselves  upon  the  basis  of  introspective  observations 
as  to  what  numerals  were  re-read  and  why  they  were  so  re-read.  In 
other  cases  definite  evidence  as  to  which  type  of  re-reading  was  used  is 
found  by  careful  examination  of  the  readings  as  they  are  described  in 
the  plates. 

The  first  type,  which  may  be  called  simple  re-reading,  has  as  its 
function,  apparently,  the  gathering  of  further  information  concerning 
the  numerals  before  a  decision  has  been  reached  as  to  what  plan  will  be 
followed  in  solving  the  problem.  Only  one  of  the  numerals  of  a  problem 
is  re-read  after  this  fashion.  A  single  case  of  exception  was  found  in 
the  reading  of  Problem  5  by  Subject  Hb  when  two  numerals  were 
re-read  in  this  way.  The  implication  is  that  such  re-reading  is  under- 
taken only  when  the  subject  is  definitely  interested  in  some  specific 
detail  of  a  certain  numeral.  According  to  the  reports  of  subjects  these 
specific  details  include  various  items  of  verification  of  numerals  such  as 
the  identity  of  certain  digits,  the  number  of  digits  in  the  numeral,  and 
the  location  of  the  numeral  within  the  line. 

The  type  of  re-reading  which  was  given  a  numeral  is  indicated  in 
Table  XXIV  in  all  cases  but  two.  The  two  exceptions  are  with  the 
readings  of  problems  2  and  3  by  Subject  M.  In  these  instances  several 
of  the  words  of  the  problems  were  re-read  along  with  the  numerals. 
Because  of  this  fact  interpretation  of  the  records  was  complicated  to 
such  an  extent  as  to  make  it  impossible  to  distinguish  with  certainty 
which  type  of  re-reading  was  given  the  numerals. 

Instances  of  simple  re-reading  are  described  in  detail  in  plates  I, 
II,  III,  V,  and  XI.  The  numeral  1000  was  given  such  a  re-reading  by 
four  of  the  subjects;  illustrations  are  found  in  plates  I  and  II.  Subject 
Hb  gave  a  re-reading  of  the  simple  type  to  numerals  in  each  of  the 
last  four  problems,  and  an  illustration  of  his  procedure  in  the  case  of 

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HOW  NUMERALS  ARE  READ 


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RE-READING  AND  COMPUTATION  71 

Problem  2  appears  in  Plate  V.  In  Plate  XI  is  found  an  instance  of  simple 
re-reading  of  the  numeral  2  along  with  certain  words  of  the  problem  by 
Subject  M. 

The  second  type  of  re-reading  of  numerals  is  that  for  the  purpose 
of  copying  the  numerals  on  to  the  problem  card.  In  the  case  of  each 
re-reading  so  classified  the  records  show  that  the  subject  after  he  had 
re-read  the  numeral,  proceeded  immediately  to  copy  it  on  the  problem 
card.  Such  re-reading  was  the  normal  procedure  for  subjects  Hb  and 
M  as  is  shown  in  Table  XXIV.  More  detailed  descriptions  of 
re-readings  of  this  type  are  found  in  plates  V,  VI,  and  VII.  The  record 
of  Subject  M,  which  is  presented  in  Plate  VI,  shows  a  distinct  variation 
from  the  procedure  of  re-reading  as  described  above.  Subject  M 
re-read  the  numeral  1,918,564  by  the  four  pauses,  24  to  27,  inclusive, 
and  at  the  same  time  copied  the  numeral  in  the  vicinity  of  pauses  32  to  34, 
without  moving  his  eyes  from  it  during  the  process  of  re-reading  and 
copying. 

Whether  a  numeral  was  or  was  not  re-read  depended  upon  the  habits 
of  individual  subjects  rather  than  upon  either  the  length  of  the  numeral 
or  upon  the  quality  of  first  reading  which  it  had  received.  Numerals 
of  all  digit-lengths,  including  both  those  which  had  received  whole  first 
readings  and  those  which  had  received  partial  first  readings,  were  re-read. 
Re-reading  was  practiced  systematically  both  by  subjects  Hb  and  M, 
the  former  of  whom  was  classified  as  a  whole  first  reader  and  the  latter 
as  a  partial  first  reader.  Of  the  four  subjects  who  almost  invariably 
proceeded  without  re-reading,  two  were  classified  as  whole  first  readers 
and  two  as  partial  first  readers. 

Attention  should  be  called  at  this  point  to  the  marked  differences 
between  the  small  proportion  of  re-readers  of  numerals,  as  reported 
immediately  above,  and  the  fact  that  all  subjects  re-read  most  of  the 
numerals  of  the  problems  which  were  used  in  the  first  preliminary  study. 
Subjects  G  and  H  who  with  only  one  exception  did  not  re-read  the 
numerals  in  the  present  study  did,  however,  persistently  re-read  the 
numerals  of  the  problems  of  the  first  preliminary  study.  In  subsequent 
paragraphs,  re-reading  is  found  to  be  closely  connected  with  the  pro- 
cedure of  copying  the  numerals  on  paper  for  computation  with  pencil. 
With  only  occasional  exceptions  this  procedure  was  followed  regularly 
by  all  of  the  subjects  in  the  first  preliminary  study,  including  subjects 
G  and  H.  It  is  probable,  therefore,  that  their  persistent  re-reading  in 
the  first  study  was  done  for  the  most  part  in  order  to  insure  accuracy 
in  copying  the  numerals  on  the  computation  paper. 


HOW  NUMERALS  ARE  READ 


2.      METHODS   AND   PROCEDURES   USED   IN   THE 
PROCESS   OF   COMPUTATION 

Two  distinct  methods  of  computation  were  exhibited  by  the  subjects 
in  respect  to  their  procedure  with  the  numerals  immediately  after  the 
first  reading  of  a  problem.  In  one  case  two  of  the  subjects  re-read  and 
copied  the  numerals  on  the  problem  card,  and  wrote  out  on  the  card 
with  pencil  the  figures  used  in  the  process  of  computation.  In  the  other 
case,  four  of  the  subjects  computed  " mentally"  and  directly  from  the 
numerals  as  they  were  printed  on  the  problem  card.  The  former  may 
be  called  computation  from  copied  figures  and  the  latter,  computation 
direct  from  the  problem  card.  Table  XXV  shows  which  procedure  was 
followed  with  each  problem  by  each  of  the  several  subjects. 

TABLE  XXV 

Two  METHODS  OF  PROCEEDING  WITH  NUMERALS  FOR  THE  PURPOSES  OF  COMPUTATION 

AFTER  THE  FlRST  READING 


Problem 

Subject 

i 

2 

3 

4 

5 

Hb 

X 

x 

x 

x 

x 

Re-read  2 

M 

o 

x 

x 

x 

x 

Re-read 

numerals 

w           

O 

o 

o 

o 

o 

H              

o 

o 

o 

o 

o 

Re-read  1000 

B  

o 

o 

o 

O 

o 

Re-read  1000 

G  

o 

0 

0 

0 

o 

Explanation  of  symbols — 

*'X"  =re-read  and  copied  numerals,  and  computed  from  copied  numerals. 
"O"=computed  immediately  and  directly  from  the  problem  card  without  re-reading. 

It  is  important  at  the  beginning  of  the  discussion  of  the  two  methods 
of  procedure  that  the  most  essential  difference  between  them  be  set 
forth  clearly.  The  difference  lies  primarily  in  the  number  of  mental 
steps  involved  in  the  two  methods.  At  least  two  additional  mental 
steps  are  required  by  the  method  of  computation  from  copied  figures, 
namely,  re-reading  of  numerals  and  copying  them  on  the  computation 
card.  The  method  of  computation  direct  from  the  problem  card  avoids 


RE-READING  AND  COMPUTATION  73 

these  two  steps.  A  large  pedagogical  significance  attaches  to  the  elimi- 
nation of  two  such  mental  steps.  By  their  elimination  it  would  appear 
that  many  opportunities  for  error  are  avoided  and  valuable  economies 
of  time  and  of  mental  and  physical  effort  may  be  effected. 

The  description  of  the  procedures  which  were  used  in  computation 
that  follows,  is  concerned  only  with  such  procedures  as  were  exhibited 
by  the  subjects  who  computed  directly  from  the  problem  cards.  It  was 
found  impossible  to  interpret  precisely  the  records  of  eye-movements 
over  copied  figures  in  respect  to  the  location  of  the  pauses  on  the  figures. 
The  processes  of  computation  could  be  followed  to  the  solution  of  the 
problem  in  only  a  limited  number  of  the  records  even  of  those  subjects 
who  used  the  method  of  direct  computation. 

The  procedure  of  direct  computation  from  the  problem  card  appears 
to  have  been  followed  without  regard  to  the  quality  of  first  reading  which 
the  numerals  had  received.  Cases  of  direct  computation  are  found  as 
sequels  both  to  partial  first  readings  and  to  whole  first  readings  of  the 
numerals.  In  Plate  XI  direct  computatiort  is  observed  to  have  followed 
upon  whole  first  readings  of  both  numerals  of  the  problem.  Similar 
illustrations  are  found  in  plates  VIII  and  X. 

In  Plate  XII,  however,  the  two  numerals  had  been  only  partially 
read  during  the  first  reading.  In  this  instance  the  numerals  never  were 
read  completely  at  any  reading.  Similar  cases  were  found  with  other 
subjects.  Evidently  for  some  subjects  only  a  partial  reading  of  the 
numerals  of  a  problem  is  necessary  for  the  successful  use  of  such  numerals 
in  computation. 

It  is  important  to  notice  in  this  connection  that  the  same  subjects 
do  not  always  use  the  same  procedure  in  computing  with  different  sets 
of  problems  under  different  conditions.  Illustrations  are  found  in  the 
cases  of  subjects  G  and  H.  These  two  subjects  used  the  method  of 
direct  computation  with  the  "five  problems,"  as  reported  immediately 
above.  With  the  "seven  problems"  of  the  first  preliminary  study, 
however,  they  followed  the  procedure  of  computation  from  copied 
figures. 

An  explanation  of  these  facts  should  begin  with  the  very  probable 
premise  that  these  two  subjects,  who  were  adults  and  advanced  graduate 
students,  were  able  to  follow  successfully  either  procedure-  in  solving 
such  simple  arithmetical  problems  as  were  used  in  this  investigation. 
The  method  of  direct  computation  was  followed  with  greater  facility 
with  the  "five  problems"  than  with  the  problems  of  the  first  preliminary 
study  because  of  the  greater  obviousness  of  the  answers  to  the  "five 


74  HOW  NUMERALS  ARE  READ 

problems."  The  numerals  of  most  of  these  problems  had  been  so 
selected  as  to  make  the  answers  even  numbers,  or  numbers  with  every 
digit  the  same,  in  order  to  minimize  the  labor  of  computation.  It  is 
probable  also  that  the  subjects  worked  at  somewhat  higher  tensions 
when  seated  before  the  camera  for  solving  the  "five  problems"  than 
when  seated  at  an  ordinary  desk  for  solving  the  "seven  problems." 
Such  higher  tensions  might  well  have  influenced  the  workers  to  select 
the  quicker  direct  method  rather  than  the  slower  copying  method  of 
solving. 

During  the  process  of  computation,  the  numerals,  which  are  in  the 
context  of  the  problem  and  with  which  the  computation  is  concerned, 
do  not  receive  the  same  quality  of  attention  from  the  subjects.  This 
fact  is  most  clearly  in  evidence  when  two  numerals  appear  in  the  context 
of  a  problem.  The  records  show  that  more  pauses  and  pauses  of  greater 
average  duration  were  located  on  the  digits  of  one  of  the  numerals  than 
on  the  digits  of  the  other  numeral.  In  the  cases  of  shorter  numerals, 
pauses  were  located  upon  only  one  of  the  numerals,  while  the  other 
numeral  was  retained  in  memory.  By  such  unequal  distribution  of 
attention  one  of  the  numerals  was,  in  effect,  made  the  "base  of  opera- 
tions" during  the  computation.  Three  examples  of  this  procedure  were 
found  in  the  solving  of  the  problems  in  which  the  shorter  numerals 
appeared.  The  records  appear  in  detail  in  plates  III,  VIII,  and  IX. 
In  Plate  IX,  which  will  serve  as  an  illustration,  the  numeral  357  was  used 
as  the  "base  of  operations, "  while  the  numeral  1643  was  neld  in  memory. 

An  example  of  the  same  procedure  with  the  longer  numerals  of 
Problem  3  is  found  in  Plate  XII.  In  this  instance  the  numeral  243,987 
was  used  as  the  "base  of  operations."  Five  pauses  with  an  average 
pause-duration  of  38/50  of  a  second  were  located  on  this  numeral  during 
the  process  of  computation.  On  the  other  hand,  on  the  numeral  21,765 
only  four  pauses  were  located  during  the  computation,  and  their  average 
duration  was  only  28.75/50  of  a  second. 

Two  further  examples  of  the  procedure  are  found  in  the  solution  of 
Problem  5  by  subjects  H  and  G  in  plates  XIII  and  XV.  Interpretation 
of  plates  XIII  and  XV  is  complicated  to  a  large  extent  by  the  presence 
in  the  records  of  a  number  of  pauses  which  were  not  used  strictly  in  the 
processes  of  computation.  Such  additional  pauses  were  used  apparently 
in  locating  the  digits  next  in  order  for  computation,  or  else  they  were 
used  in  directing  the  hand  when  it  was  engaged  in  recording  figures  of 
the  answer.  Such  pauses,  therefore,  may  be  referred  to  as  locating- 
and  as  recording-pauses  respectively. 


RE-READING  AND  COMPUTATION 


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76  HOW  NUMERALS  ARE  READ 

It  was  found  impossible  to  distinguish  the  locating-  and  recording- 
pauses  in  plates  XIII  and  XV  from  the  computation  pauses  with  abso- 
lute certainty.  An  effort  was  made,  however,  after  a  detailed  study  of 
the  original  reports  of  subjects  H  and  G  on  their  solutions  of  Problem  5, 
to  separate  the  locating-  and  recording-pauses  from  the  computation 
pauses.  The  result  appears  in  Table  XXVI.  An  inspection  of  the  table 
shows  that  both  subjects  H  and  G  used  the  numeral  1,918,564  as  the 
"base  of  operations"  during  computation. 

The  longer  numeral  in  five  of  the  six  cases,  which  are  described  above, 
was  taken  as  the  "base  of  operation."  Such  selection  of  the  longer 
numeral  is  probably  in  keeping  with  the  practice  common  in  the  solving 
of  arithmetical  problems  which,  when  two  numerals  are  arranged  for 
computation,  places  the  greater  numeral  first  in  order  and  relates  the 
second  numeral  to  the  greater.  The  larger  number  of  pauses  upon  the 
longer  numeral,  which  at  the  same  time  is  the  "base  of  operations," 
is  due  to  the  fact  that  computation  both  begins  and  ends  with  the  longer 
numeral,  and  to  the  further  fact  that  in  the  longer  numeral  an  additional 
digit  appears  for  reading. 

The  large  difference  between  the  average  duration  of  the  pauses  on 
the  numeral  which  was  used  as  the  "base  of  operations"  and  the  average 
duration  of  the  pauses  on  the  other  numeral  is  significant  of  a  difference 
in  function  between  the  two  sets  of  pauses.  The  pauses  on  both  numerals 
necessarily  must  use  such  time  as  is  sufficient  for  recognition  on  the  part 
of  the  reader  of  the  digits  with  which  the  pauses  are  severally  concerned. 
In  addition  to  the  work  of  recognitions,  however,  some  of  the  pauses 
on  the  "base  of  operations"  numeral  evidently  perform  service  in  the 
more  strictly  arithmetical  processes.  For  such  service  a  greater  pause- 
duration  would  undoubtedly  be  necessary. 

3.      SUMMARY 

1.  Two  types  of  re-reading  of  numerals  are  distinguishable.     Simple 
re-reading  is  concerned  with  verification  of  details  of  the  numerals. 
Re-reading  for  copying  is  concerned  with  reading   the  numerals  for 
copying  on  the  computation  card. 

2.  Two  of  the  subjects  normally  re-read  all  of  the  numerals.     The 
four  other  subjects  normally  do  not  re-read  the  numerals.     Whether 
the  numerals  are  or  are  not  re-read  depends  upon  the  habits  of  individual 
subjects. 

3.  Two  methods  of  proceeding  with  the  numerals  after  the  first 
reading  are  distinguishable.     In  the  one,  computation  begins  immedi- 


RE-READING  AND  COMPUTATION  77 

ately  and  is  carried  on  directly  from  the  context  of  the  problem.  In  the 
other  case  the  numerals  are  re-read  and  copied,  and  the  computation 
proceeds  from  the  copied  figures.  By  the  former  method,  two  mental 
steps  are  saved. 

4.  The  method  of  immediate  computation  direct  from  the  context  of 
the  problem  is  used  without  regard  to  whether  the  numerals  have 
received  a  partial  or  a  whole  first  reading. 

5.  During  computation  one  numeral  is  taken  as  the  "base  of  opera- 
tions."    A  large  number  of  pauses  and  pauses  of  greater  average  duration 
are  located  on  the  digits  of  this  numeral  than  on  the  digits  of  the  other 
numeral.     The   significance   of   the  greater  duration  of  such  pauses 
probably  lies  in  the  additional  work  of  the  more  strictly  arithmetical 
processes  which  seems  to  have  been  done  during  these  pauses. 


CHAPTER  IX 

THE  READING  OF  ISOLATED  NUMERALS  IN  LINES 
I.      INTRODUCTION 

As  was  stated  in  the  introductory  paragraphs  of  this  report  such 
attention  as  has  been  given  to  the  reading  of  numerals  in  previous 
experiments  in  the  psychology  of  reading  has  been  incidental  to  other 
purposes.  The  numerals  which  were  read  in  previous  experiments 
were  in  each  case  isolated  numerals  in  lines.  Gray,1  when  investigating 
the  perception  span  of  good  and  poor  readers,  had  a  number  of  individuals 
read  short  lines  of  unspaced  digits  and  groups  of  the  same  digit  as  well 
as  selections  of  words  with  meaning.  A  summarizing  paragraph  at  the 
end  of  his  discussion  contains  the  conclusion  that  differences  between 
the  span  of  attention  of  the  good  and  poor  reader  disappear  in  a  very 
large  measure  when  digits  or  groups  of  the  same  digit  are  read. 

Dearborn,2  while  interested  chiefly  in  the  span  of  attention  and  in 
the  question  as  to  whether  perception  proceeds  by  number  wholes  or  by 
individual  digits,  had  several  subjects  read  lines  of  digits  which  were 
printed  consecutively  without  spacing,  and  lines  of  numerals  varying 
in  length  from  two  to  six  digits.  In  the  records  which  were  obtained 
from  these  readings  he  observed  that  the  time  required  for  reading  the 
same  number  of  unspaced  digits  in  a  line  was  greater  when  the  subjects 
grouped  the  digits  by  fours  than  when  they  grouped  the  digits  by  threes. 
He  noticed  also  that  the  time  required  for  reading  numerals  increased 
with  increases  in  the  digit-lengths  of  the  numerals.  Attention  was 
called,  at  the  same  time,  to  the  larger  number  of  " shifts"  or  "breaks" 
in  the  fixations  on  numerals  than  in  the  fixations  on  words,  and  the 
opinion  was  expressed  that  a  single  digit  was  probably  sometimes  the 
unit  of  perception  in  the  two-digit  numerals. 

In  a  preliminary  study  of  the  present  investigation,  data  were 
presented  concerning  the  reading  of  numerals  arranged  in  columns. 

1  C.  T.   Gray,  "Types   of   Reading    Ability  as  Exhibited  through  Tests  and 
Laboratory   Experiments,"   Supplementary  Educational  Monographs,  Vol.  I,  No.  5 
(1917),  p.  146. 

2  W.  F.  Dearborn,    "Psychology  of   Reading:    An  Experimental  Study  of  the 
Reading  Pauses  and  Movements  of  the  Eye,"  Columbia  University  Contributions  to 
Philosophy  and  Psychology,  Vol.  XIV,  No.  i.     New  York:   The  Science  Press,  1906. 
See  chapter  x,  "The  Number  Span  of  Attention,"  pp.  67-73. 

78 


THE  READING  OF  ISOLATED  NUMERALS  IN  LINES  79 

In  that  study  special  attention  was  called  to  the  fact  that  the  digits 
of  the  numerals  were  read  in  groups.  The  purpose  of  the  present  study 
is  to  examine  in  detail  the  readings  of  a  representative  number  of  numerals 
of  each  of  the  several  digit-lengths  of  from  one  to  seven  digits.  Varia- 
tions in  the  readings  of  the  numerals  of  the  several  lengths  are  reported 
and  the  reading  habits,  which  were  exhibited  by  individual  readers, 
are  described.  The  records  of  the  movements  of  the  eyes  of  the  subjects 
while  engaged  in  reading  numerals  are  given  in  plates  XVI-XXV,  and 
the  data  from  the  records  are  condensed  and  [arranged  in  tables  XXVII- 
XXXI. 

In  the  plates  the  variations  in  length  of  the  vertical  lines  above  and 
below  the  lines  of  printed  numerals  were  provided  merely  for  convenience 
in  drawing  in  the  numbers  of  the  various  pauses. 

When  the  initial  pause  of  a  line  did  not  fall  on  one  of  the  digits  of 
the  first  numeral  in  the  line,  such  a  pause  was  not  included  in  the  tables. 
When  an  initial  pause  fell  on  the  first  numeral  of  a  line  and  was  followed 
by  a  regressive  movement,  such  a  pause  was  not  counted  in  the  tables. 
It  is  obvious  that  counting  pauses  of  the  latter  sort  would  have  given 
in  each  case  an  additional  pause  to  the  first  numeral  in  the  line  merely 
because  of  its  position  as  first  numeral  in  the  line. 

In  plates  XVI  and  XVII  the  reading  of  isolated  numerals  by  Subject 
G  is  represented.  At  the  beginning  of  each  line  of  numerals  an  initial 
regressive  movement  was  found  necessary  in  the  effort  to  locate  the  first 
digits  of  the  first  numeral.  Subject  G  reads  with  relatively  few  pauses, 
but  with  pauses  of  relatively  long  duration.  The  pauses  vary  widely 
in  duration;  the  range  of  variation  extends  from  4/50  to  90/50  seconds 
with  the  average  duration  at  33.88/50  of  a  second.  Single  pauses,  when 
they  are  located  on  the  longer  numerals,  tend  to  perceive  two  or  three 
digits  rather  than  one  or  two.  In  three  instances  the  subject 
accomplished  the  remarkable  feat  of  reading  four  digits  during  a  single 
pause.  The  three  instances  are  found  in  Plate  XVII;  two  are  in  the 
first  line  with  the  numerals  9317  and  5,236,795;  and  the  third  is  in  the 
second  line  with  the  numeral  1928. 

In  plates  XVIII  and  XIX  appear  the  readings  of  isolated  numerals 
by  Subject  H.  This  subject  read  with  a  relatively  large  number  of 
pauses,  but  with  pauses  of  relatively  short  duration.  The  pauses 
varied  in  duration  from  4/50  to  56/50  seconds.  The  average  dura- 
tion, which  was  19.32/50  of  a  second,  was  shorter  than  that  of  any  other 
subject.  Many  short  guiding-pauses  appear.  Single  pauses,  even 
when  they  are  located  on  the  longer  numerals,  tend  to  include  only 


8o 


HOW  NUMERALS  ARE  READ 


836 

fcs  7 


1     )i 


4,      ,5 

9 

28"      "29 


PLATE  XVI 

v 

756, 


*,9F4 

4,2         21  k 


24    33 


31         90 


33    15 


VI 

354,908 

42      17 


J2  4441  Tl°  II 

74  30    29   16  8  14.      II  24      I: 


9.    ,10  <e 


1 

2J 

30    29    16  8  J      II  24     -13  32       9 

Reading  of  isolated  numerals  by  Subject  G 


76 


59 


'I  I 

9317 

T  «V 


PLATE  XVII 

6. 


IF 

25 


\  1        1 

,236,7*5  26( 

Js  Jg.  J, 


il 
748I 


3          «. 

9 


38      33 


365 

35 


7.       8.  y.      10 


:,548 

47       40 


i98l,6* 

31       28       29 


3       II 


TIT 

?9  2t      «     34 

Reading  of  isolated  numerals  by  Subject  G 


4 

54,  fr  14' 


THE  READING  OF  ISOLATED  NUMERALS  IN  LINES  81 

PLATE  XVIII 

IP    ||    9      IZ  !3 


'*    2.1 

3313 


I  J*      i 

85,914  2$c 

45        32  45 


ttppg  ip2tt  L 

3    \X    U  I  35        26 


U.      i         13 


490p 

4/        £0     35 


£4 


2 


IT  'I 

TT 

&   Ji<*       7 


l'9[l  IT  I 


43   43 "10  25     9  48  20'3| 

Reading  of  isolated  numerals  by  Subject  H 

PLATE  XIX 
3.  '°.     «« 


5  /o       u  IP       13 

J7  fe  6,2:16, 

U  «4         5  8^      ^ 


15 


'*  9'   38'M 


4         I 


12 1    «T 

IT     T 


12. 


\L 

\\ 

£4       * 


tS 


I 

T 


' 


es  «3  «» 


687, 


Reading  of  isolated  numerals  by  Subject  H 


(2.    13 


82 


3.     ?•  i 


HOW  NUMERALS  ARE  READ 
PLATE  XX 

\Z     13     9 


4-    5 


33  fe  33 


33     5     J*    B 


'24 


S    V 


Y  i'   'i  r     i     'i  i  f  >3  T  f     1 

1      fl        1        ftt        H        I 

5   T      r  -ii     7  10  3f   23  13         4  15     24  24 


If.     /A    .1?  .19 

& 


40 


}  }ft 


.  1  T 

Reading  of  isolated  numerals  by  Subject  M 
PLATE  XXI 


},i|4  Jsir  ip  I  sbbjAJ 

el        4  29       «4  'l9  T      T  «•  I      •'   4  " 


14 


f 


2< 

I 


15 


to  »3   a  3 


!'  M 

«         u1  'e 


14 


• 


Reading  of  isolated  numerals  by  Subject  M 


THE  READING  OF  ISOLATED  NUMERALS  IN  LINES  83 


1 

T 

50       5 


PLATE  XXII 

3     T    .10 


TT 

32      31 


TT 


31     24  '*    4' 


36, 


20    15       t>3        17 


"llf 

8,7 

55    44    8  r          " 


33       44 


21 


14       IS 


9?      13      27 


I     'Ultf        I         9J 

T         rf3!4'?5!          T 

&T  ?     309   4»     JiW  L 


51  7        40 

Reading  of  isolated  numerals  by  Subject  B 

PLATE  XXIII 


* 1 '    HT      1      I 

7618tt  pplH  J7  B 

2o     34      (o  10  41  it  T  3T  fc* 


21       30       51 


53 


51 


3 
5     I 


12    14    J5 .13  |6      IT 


f        39     35 


43 


f41 


26  14  34  33      3o -Jl 

Reading  of  isolated  numerals  by  Subject  B 


84 


HOW  NUMERALS  ARE  READ 
PLATE  XXIV 


2  I         5 


1 3 


,4      15     »(*      IT 


IS     5>5  23  T       47  W  »   73 


756,35! 

4*       35 


:,B2B,9*6 

JO     21    -2i    39 


>H  I 

'3      32   4  to  23  3k  « 


5,   H,  'V°  '3,     ,'4  IS,     »6    '.T 


15 


48    ;H 

if     40  2T  T  42.     11 


354,  ) 

5?       3^    34- 


1 


H" 


12, 


25        S  19      3k  /9     'I  t&     ^2  9     43  24 

Reading  of  isolated  numerals  by  Subject  W 


PLATE  XXV 


38      34 


frn     14 


33  38 


31  19      40     21      37 


•  •  ** 

45     T52T? 


H 


10       .11  12.  13.15    .14    16 


30 


IT  r  i 

,8467 
it  af§  'i*  2 


e,  ^  t,« 


45 


25'    '3  a«  13      14  23   20  31   T4  ^ 

Reading  of  isolated  numerals  by  Subject  W 


THE  READING  OF  ISOLATED  NUMERALS  IN  LINES  85 

one  or  two  digits  at  a  pause.  The  readings,  which  were  given  the 
numerals  756,352  in  Plate  XVIII  and  93,548  in  Plate  XIX,  are  illustra- 
tions of  very  detailed  reading.  The  readings  given  the  last  three 
numerals  on  Plate  XVIII,  the  special  numerals  1000,  333,  and  25,000 
show  fewer  pauses  than  H  gave  to  other  numerals  of  like  lengths.  He 
evidently  read  1000  as  a  whole.  A  relatively  large  number  of  regressive 
pauses  is  found  on  these  plates. 

The  readings  of  Subject  M  are  described  in  plates  XX  and  XXI. 
His  methods  of  reading  the  numerals  were  similar  to  those  of  Subject  H, 
to  which  attention  was  called  in  connection  with  plates  XVIII  and  XIX. 
The  total  reading- time  for  all  the  numerals  was  less  for  Subject  M  than 
for  any  other  subject,  despite  the  relatively  large  number  of  pauses 
which  he  used.  Several  instances  appear  in  these  plates  of  short  initial 
and  final  pauses  on  the  same  longer  numerals.  The  readings  of  the 
numerals  4,325,986  and  16,789  in  Plate  XX  and  of  5,236,795  and  743,819 
in  Plate  XXI  illustrate  such  use  of  the  guiding-pauses.  The  special 
numerals  were  read  in  the  same  manner  as  other  numerals  of  like  lengths. 
Plates  XXII  and  XXIII  record  the  reading  of  isolated  numerals  by 
Subject  B.  It  appears  that  this  subject's  readings  are  irregular  in 
respect  to  number  of  pauses  on  numerals  of  the  greater  lengths.  Some 
of  the  longer  numerals  were  read  with  few  pauses.  The  numerals 
5,236,795  and  3,984,673  in  Plate  XXIII  are  illustrations  of  this  type, 
while  the  numerals  9317  and  743,819  on  the  same  plate  were  read  with  a 
comparatively  large  number  of  pauses.  The  numeral  1000  was  evidently 
read  as  a  whole.  Initial  regressions  appear  consistently  in  each  line. 

The  readings  of  isolated  numerals  by  Subject  W  are  presented  in 
plates  XXIV  and  XXV.  A  persistent  use  of  two  pauses  appears  in 
the  readings  of  numerals  of  from  three-  to  five-digit  lengths.  Two 
instances  are  seen  in  the  numerals  5489  and  16,789  in  Plate  XXIV  when 
even  the  re-readings  of  the  numerals  were  done  with  pairs  of  pauses. 
Initial  regressions  occur  consistently. 

2.      TWO   TYPES   OF   PAUSES 

When  a  detailed  examination  is  made  of  the  pauses  with  which  the 
numerals  were  read,  it  appears  that  they  represent  two  distinct  types. 

The  two  types  are  distinguished  by  differences  in  function.  Pauses 
of  the  first  type,  which  may  be  called  strictly  reading-pauses,  were 
probably  used  in  recognizing  the  identity  of  the  digits  of  the  numerals 
and  the  relations  between  the  digits.  Such  pauses  are  invariably  located 
on  the  numerals.  Their  durations  are  approximately  equal  to,  or  greater 


86  HOW  NUMERALS  ARE  READ 

than,  the  average  duration  of  the  pauses  of  the  subject  whose  records 
are  under  consideration.  A  preponderant  number  of  the  pauses  of  any 
subject  are  of  this  first  type. 

Pauses  of  the  second  type,  which  may  be  called  guiding-pauses, 
were  probably  used  in  locating  the  first  digits  or  the  last  digits  of  the 
numerals.  They  are  found  on  the  initial  or  final  digits  of  numerals, 
and  more  frequently  on  numerals  of  greater  digit-lengths.  Some  of 
these  pauses  appear  on  the  lines  between  the  numerals.  The  first  pause 
in  any  line  was  almost  invariably  of  this  type  and  of  very  brief  duration 
when  compared  with  other  pauses.  Subjects  H  and  M  used  larger 
numbers  of  pauses  of  this  type  than  any  of  the  other  subjects. 

3.      DIFFERENCES   IN   THE   READINGS  OF  NUMERALS    OF 
DIFFERENT  LENGTHS 

Upon  inspection  of  the  last  row  of  Table  XXVII  it  is  found  that  the 
average  total  reading-time  per  numeral  increases  steadily  from  an 
average  of  21.45/50  of  a  second  for  the  one-digit  numerals  to  an  average 
of  104.54/50  of  a  second  for  the  seven-digit  numerals.  The  same 
continual  increase  is  found  almost  without  exception  in  the  rows  of  the 
several  subjects.  Likewise  the  average  number  of  pauses  per  numeral, 
when  the  records  of  all  subjects  are  averaged,  increases  steadily  from 
the  average  of  1.15  pauses  on  one-digit  numerals  to  the  average  of  4.15 
pauses  on  seven-digit  numerals.  It  is  clear  therefore  that  the  total 
reading-times  and  the  number  of  pauses  which  were  required  to  read 
a  numeral,  depended  upon  its  digit-length. 

The  average  duration  of  the  pauses,  on  the  other  hand,  does  not 
depend  upon  the  length  of  the  several  numerals  in  the  same  consistent 
fashion  as  does  the  average  number  of  pauses  per  numeral.  The  details 
may  be  found  in  Table  XXVIII,  where  it  is  seen  that  both  in  the  rows 
for  individual  subjects  as  well  as  in  the  row  which  presents  averages  for 
all  of  the  subjects,  the  average  pause-duration  not  only  fails  to  increase 
steadily  with  increasing  lengths  of  the  numerals  but  in  several  cases 
actually  decreases. 

With  three  of  the  subjects,  M,  H,  and  G,  on  the  other  hand,  the 
average  duration  of  the  pauses  increases  steadily  from  that  of  the 
numerals  of  one  digit  to  that  of  the  numerals  of  three  digits.  Two  of 
these  three  subjects,  M  and  G,  however,  as  may  be  observed  in  Table 
XXIX,  read  the  one,  two,  and  three  digits  of  the  one-,  two-,  and  three- 
digit  numerals,  respectively,  for  the  most  part  with  a  single  pause. 
In  the  case  of  Subject  B,  also,  a  steady  increase  in  pause-duration  is 
found  when  the  one-,  two-,  and  three-digit  numerals  were  read  at  single 


THE  READING  OF  ISOLATED  NUMERALS  IN  LINES 


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HOW  NUMERALS  ARE  READ 


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THE  READING  OF  ISOLATED  NUMERALS  IN  LINES 


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HOW  NUMERALS  ARE  READ 


pauses.  It  is  evident,  therefore,  that  with  certain  subjects  and  to  this 
limited  extent,  the  pause-duration  does  increase  with  increases  in  the 
number  of  digits  read. 

Numerals  of  the  same  length  exhibited  a  notable  consistency  in  the 
number  of  pauses  with  which  they  were  read  by  the  several  subjects. 
As  may  be  observed  in  Table  XXIX  two  different  numbers  of  pauses 
include  the  number  of  pauses  used  in  reading  a  preponderant  number  of 
the  numerals  of  each  length.  The  one-  and  two-digit  numerals  were 
read  for  the  most  part  by  single  pauses.  One  and  two  pauses  were 
used  with  the  three-digit  numerals  and  two  and  three  pauses  with  the 
four-  and  five-digit  numerals  in  most  cases.  The  longer  numerals 
exhibit  more  variety  in  the  number  of  pauses  with  which  they  were 
read.  With  these  numerals,  the  habits  of  individuals  seem  larger 
deciding  factors  than  the  tendency  to  a  conventional  number  of  pauses. 

4.      THE   SPECIAL  NUMERALS 

The  six  special  numerals,  o,  99,  333,  1000,  25,000,  and  637,637,  were 
included  among  the  other  numerals,  in  order  that  data  might  be  obtained 
concerning  such  variations  in  the  readings  of  numerals,  as  might  be  due 
to  familiarity  and  to  regularity  in  form  on  the  part  of  the  numerals 
read.  The  data  for  the  two  special  numerals  1000  and  25,000  show 
consistent  and  significant  variations  from  the  data  which  represent  the 
readings  of  ordinary  numerals  of  like  lengths.  As  may  be  observed  in 
Table  XXX  the  two  numerals  1000  and  25,000  exhibit  markedly  shorter 
average  reading- time  per  numeral,  average  number  of  pauses  per  numeral, 
and  average  pause-durations,  than  the  ordinary  numerals  of  the  same 
lengths  with  which  they  are  compared. 

TABLE  XXX 

READING  OF  SPECIAL  NUMERALS  COMPARED  WITH  READING  OF  OTHER  ISOLATED 

NUMERALS  OF  CORRESPONDING  DIGIT-LENGTHS 

(Time  unit  =  1/50  of  a  second) 


Special  numerals  

o 

99 

333 

IOOO 

25  ,OOO 

637  ,637 

Corresponding  digit-lengths  of  other  isolated  numer- 
als with  which  the  special  numerals  are  compared 

i 

2 

3 

4 

5 

6 

Average  reading  -time  per  numeral: 
Special  numerals  

16.4 

30.8 

46.6 

33.6 

53.8 

93  -O 

Other  numerals 

21  45 

28  7 

48  05 

59  "5 

77  75 

94  OS 

Average  number  of  pauses  per  numeral: 
Special  numerals 

I    O 

i  40 

I  40 

i  6 

2.6 

4.  50 

Other  numerals  

i  •  15 

1  .  2 

1  .9 

2  .4 

2.9 

3-7 

Average  pause-duration: 
Special  numerals  

16.4 

22.  0 

33-29 

21.0 

24.46 

25.82 

Other  numerals     

19.3 

20.86 

27  .04 

26.28 

28.43 

26.88 

NOTE. — Each  subject  read  each  special  numeral  once,  and  at  the  same  reading  with  the  other  isolated 
numerals. 


THE  READING  OF  ISOLATED  NUMERALS  IN  LINES  91 

Each  of  the  two  numerals  represents  the  special  quality  of  regularity 
of  form  and  the  quality  of  familiarity  which  were  described  in  the 
discussion  of  the  numeral  1000  in  chapter  vii.  Regularity  of  form,  in 
what  may  be  called  common  fashion  as  distinguished  from  the  special 
fashion  of  regularity  which  has  been  described  for  the  numeral  1000, 
is  a  quality  of  the  other  special  numerals  99,  333,  and  637,637.  Such 
regularity  of  form,  however,  did  not  seem  to  be  significant  enough  to 
affect  important  variations  in  the  readings  of  those  numerals.  This 
immediate  fact  and  the  facts  which  are  noted  above  appear  in 
re-enforcement  of  the  conclusion  reached  regarding  the  numeral  1000 
in  chapter  vii.  Evidently  the  qualities  of  special  regularity  of  form  and 
familiarity  are  influential  factors  in  the  reading  of  numerals,  even  when 
the  numerals  are  isolated  in  lines. 


5.      TWO   METHODS   OF   ATTACK   IN   READING   NUMERALS 

Two  distinct  methods  of  attack  were  used  by  different  subjects  in 
reading  the  numerals.  By  one  method  a  relatively  large  number  of 
pauses  of  relatively  short  average  duration  were  used  in  the  readings, 
and  by  the  other  relatively  few  pauses  of  relatively  long  average  duration 
were  used.  The  contrast  between  the  two  methods  is  sharply  outlined 
by  differences  in  the  data  for  subjects  H  and  G,  who  represent  the  first 
and  second  methods  respectively,  as  such  data  are  arranged  in  Table 
XXXI.  Subject  M  also  read  by  the  first  method  and  subjects  W  and 
B  by  the  second  method.  In  plates  XVIII  and  XIX  a  relatively  large 
number  of  guiding-pauses  are  found,  while  comparatively  few  appear 
in  plates  XVI  and  XVII.  The  shortest  total  reading-times  for  the  whole 

TABLE  XXXI 

SPEED  AND  THE  Two  METHODS  OF  ATTACK  USED  IN  READING  ISOLATED  NUMERALS 
(Time  unit  =  1/50  of  a  second) 


Methods  of  attack 

Many  Pauses  of 
Short  Average 
Duration 

Fewer  Pauses  of  Longer 
Average  Duration 

Subjects    

M 

H 

G 

W 

B 

Total  reading-time  for  all  nu- 
merals   

J549 

76 
20.38 

1642 

85 
19.32 

1728 
33-38 

1797 

69 
26.04 

1937 

67 
28.92 

Total  number  of  pauses  used  in 
reading  all  numerals  . 

Average  pause-duration  

NOTE. — Each  subject  read  four  each  of  the  one-,  two-,  three- and  seven-digit  numerals    or 

a  total  of  twenty-eight  numerals. 


92  HOW  NUMERALS  ARE  READ 

set  of  isolated  numerals  are  found  in  the  records  of  subjects  M  and  H 
who  used  the  many-short-pauses  method  of  attack  in  their  readings. 

The  number  of  digits  which  are  read  at  a  single  pause  depends  upon 
the  method  of  attack  which  the  subject  uses,  and  upon  the  number  of 
digits  which  a  numeral  offers  for  reading.  Subjects  M  and  H,  who  use 
the  method  of  many  short  pauses,  tend  to  perceive  one  and  two  digits 
at  a  pause.  Subjects  G,  B,  and  W,  on  the  other  hand,  who  use  the 
method  of  few  and  long  pauses,  tend  to  perceive  two  and  three  digits 
at  a  pause.  The  one  and  two  digits  of  the  one-  and  two-digit  numerals 
were  almost  invariably  read  by  single  pauses  by  all  subjects.  In  all 
of  the  plates  which  describe  the  readings  of  M,  G,  W,  and  B  instances 
are  to  be  found  among  the  readings  of  the  longer  numerals  in  which  as 
many  as  three  digits  were  read  at  a  pause. 

In  only  three  instances,  however,  when  the  special  form  numeral 
1000  is  not  brought  into  consideration,  as  many  as  four  digits  are  known 
to  have  been  read  at  a  single  pause.  All  three  of  the  instances  are  in 
the  readings  of  Subject  G  and  are  reported  on  Plate  XVII.  Two 
instances  occur  on  the  four-digit  numerals  9317  and  1928,  and  the  other 
on  the  seven-digit  numeral,  5,236,795.  Evidently  the  reading  of  four 
digits  at  a  single  pause  is  very  exceptional  and  it  is  significant  that  it 
occurs  only  in  the  records  of  the  subject,  who  used  the  smallest  number 
of  pauses  with  the  greatest  average  duration. 

6.     SUMMARY 

1.  Two  types  of  pauses  are  distinguished  by  differences  in  duration. 
The  strictly  reading-pauses  are  probably  used  to  recognize  the  identity 
of  the  digits  of  the  numerals,  and  the  relations  between  them.     The 
guiding-pauses  are  used  probably  in  locating  the  initial  and  final  digits 
of  numerals. 

2.  The  average  total  reading- time  per  numeral  and  the  average 
number  of  pauses  per  numeral  increase  gradually  from  the  averages  of 
the  one-digit  numerals  to  the  averages  of  the  seven-digit  numerals. 

3.  With  three  subjects  the  average  duration  of  pauses  increased 
with  increases  in  the  number  of  digits  read  by  the  pauses. 

4.  Two  different  numbers  of  pauses  include  the  number  of  pauses 
used  in  reading  a  preponderant  number  of  the  numerals  of  each  length. 
The  greatest  consistency  in  this  respect  is  found  in  the  numerals  of  from 
one  to  five  digits  in  length. 

5.  The  quality  of  familiarity,  rather  than  the  quality  of  regularity 
of  form  reduced  the  average  number  of  pauses  and  the  total  reading- 


THE  READING  OF  ISOLATED  NUMERALS  IN  LINES  93 

times  of  the  numerals  1000  and  25,000  below  the  averages  of  ordinary 
numerals  of  like  lengths. 

6.  Two  distinct  methods  of  attack  are  used  in  reading  numerals. 
By  the  one  method  a  relatively  large  number  of  pauses  of  relatively 
short  average  duration  is  used,  and  by  the  other  method  relatively 
few  pauses  of  relatively  long  average  duration  are  used.     The  subjects 
who  employed  the  former  method  read  the  numerals  in  shorter  total 
reading-time. 

7.  The  usual  number  of  digits  read  per  pause  is  one  or  two,  or  two 
or  three  according  to  the  habits  of  the  subject  who  is  reading.     One 
subject  is  able  to  read  as  many  as  four  digits  at  single  pauses. 


CHAPTER  X 
COMPARISONS  OF  RATES  OF  READING 

I.      COMPARISON   OF   THE    SUBJECTS    OF   THE   PRESENT   INVESTIGATION 

WITH   THE    SUBJECTS    OF   SCHMIDT'S    INVESTIGATION   IN 

RESPECT  TO   RATES   OF   READING 

Attention  has  been  called  in  previous  sections  of  this  report  to  the 
fact  that  larger  numbers  of  pauses  of  relatively  greater  average  duration 
were  used  in  reading  the  arithmetical  problems  than  are  commonly 
used  in  reading  ordinary  prose  materials.  It  was  decided,  therefore, 
to  test  the  reading-speeds  of  the  several  subjects  of  the  present  investiga- 
tion with  a  different  type  of  reading-material.  The  data,  which  were 
thus  obtained,  could  then  be  compared  with  other  data  which  represented 
he  reading-speeds  of  similar  individuals  with  similar  materials. 

The  type  of  material,  which  was  selected  as  most  appropriate  for 
this  purpose,  was  that  of  ordinary  expository  prose.  The  text  of  the 
selection  was  taken  from  Judd's  Psychology  of  High-School  Subjects, 
a  volume  which  was  familiar  in  a  general  way  to  all  of  the  readers. 
The  data  obtained  with  this  material  were  to  be  compared  with  the 
results  reported  by  Schmidt1  in  a  previous  investigation  of  the  reading 
of  "light  passages  from  James's  Psychology"2  by  45  "adults,  mostly 
graduate  and  undergraduate  students."3 

The  conditions  which  governed  the  readings  of  the  two  selections 
were  similar  for  the  most  part.  In  one  respect,  however,  an  impor- 
tant difference  obtained.  In  Schmidt's  investigation  the  subjects  were 
instructed  to  "read  rapidly  for  the  thought."4  In  the  present  investi- 
gation, out  of  deference  to  purposes  which  are  stated  in  the  latter  part  of 
this  chapter,  the  subjects  were  instructed  to  read  for  a  "  clear  understand- 
ing" and  at  "normal  speed."  Further  details  concerning  this  material 
and  the  conditions  under  which  it  was  given  appear  in  chapter  vi. 

In  order  to  facilitate  the  comparison  certain  rearrangements  of  the 
data  as  originally  reported  by  Schmidt  were  made.  The  time  unit 
used  in  his  investigation  was  the  sigma,  or  i/iooo  of  a  second.  Figures 

1  W.  A.  Schmidt,   "An   Experimental   Study  in  the  Psychology  of  Reading,"" 
Supplementary  Educational  Monographs,  Vol.  I,  No.  2  (1917),  p.  42. 

2  Ibid.,  pp.  32.  50.  ^Ibid.j  p.  34.  *  Ibid.,  p.  28. 

94 


COMPARISONS  OF  RATES  OF  READING 


95 


which  represented  duration  of  time  in  this  unit  were  converted  into  other 
figures  representing  the  same  durations  in  units  of  1/50  of  a  second. 
By  multiplying  average  pause-duration  by  average  number  of  pauses 
per  line  figures  were  obtained  for  the  average  reading-time  per  line. 
In  this  way  a  single  number  is  found  to  represent  rates  of  reading. 
The  data  from  the  two  investigations  are  arranged  in  Table  XXXII. 

When  the  data  in  Table  XXXII  are  compared  it  is  found  that  the 
subjects  of  the  present  investigation  read  at  conspicuously  higher  rates, 
as  judged  by  average  reading-time  per  line,  than  the  subjects  of  Schmidt's 
investigation.  The  superiority  in  speed  on  the  part  of  the  former 
subjects  is  due  in  largest  measure  to  the  decidedly  shorter  average  dura- 
tions of  their  pauses,  although  they  also  used  fewer  pauses  per  line. 

TABLE  XXXII 

SUBJECTS  OF  THE  PRESENT  INVESTIGATION  COMPARED  WITH  THOSE  OF  SCHMIDT'S 

INVESTIGATION  IN  RESPECT  TO  SPEED  OF  READING 

(Time  unit  =  1/50  of  a  second) 


SUBJECTS 

Average 
Number 
of  Pauses 
per  Line* 

Average 
Pause- 
Duration 

Average 
Deviation 

Average 
Reading- 
Time 
per  Linet 

Of  the  present  investigation  (5  adults) 
Of  Schmidt's  investigation  (45  adults) 

6.05 
6.50 

10-75 
!5-4it 

2.87 
3-93 

65.04 
100.  17 

*The  line-length  of  the  materials  read  by  Schmidt's  subjects  was  90  mm.;  that  of  the  materials 
read  by  the  subjects  of  the  present  investigation  was  93  mm. 

t  Schmidt's  results,  which  are  reported  by  him  in  time  units  of  sigma  (i/iooo  of  a  second),  are  here 
presented  in  time  units  of  1/50  of  a  second. 

J  The  average  reading-time  per  line  is  obtained  by  multiplying  the  average  number  of  pauses  per 
line  by  the  average  pause-duration. 

The  advantage  in  favor  of  the  subjects  of  the  present  investigation  is 
emphasized  by  the  fact  that  they  were  reading  under  instructions  which 
called  for  only  " normal  speed,"  whereas  the  other  subjects  read  under 
the  instructions,  "read  rapidly  for  the  thought." 

So  large  a  number  of  adult  cases  was  included  in  the  investigation 
by  Schmidt  that  the  figures  reported  by  him  may  be  taken  as  representing 
reliable  averages  of  the  performances  of  such  subjects.  Upon  the  basis 
of  comparison  with  the  results  thus  reported  it  is  concluded  that  the 
subjects  of  the  present  investigation  may  be  classified  as  decidedly 
better  than  average  adults  in  respect  to  speed  of  reading.  The  implica- 
tion of  the  foregoing  paragraphs  and  of  this  conclusion  is  obvious. 
The  larger  number  of  pauses  of  relatively  greater  duration,  which  was 


96 


HOW  NUMERALS  ARE  READ 


found  in  the  readings  of  the  problems,  is  due  to  the  nature  of  the  problems 
as  a  type  of  reading-material  rather  than  to  slow  speeds  on  the  part  of 
the  readers. 


2. 


COMPARISONS    OF   RATES   OF   READING   THE    THREE    TYPES   OF 
READING-MATERIALS 


Significant  evidence  of  differences  between  types  of  reading-materials, 
and  concerning  the  nature  of  the  differences  as  well,  may  be  secured  by 
comparisons  between  the  rates  with  which  the  subjects  read  the  different 
types  of  materials.  With  such  a  purpose  in  view,  comparisons  were 
arranged  between  the  data  from  the  readings  of  the  problems,  of  the 
isolated  numerals,  and  of  the  ordinary  expository  prose  selection.  The 
conditions  which  governed  the  readings  of  all  three  types  of  materials 
were  essentially  similar  in  that  they  called  for  such  kinds  of  reading  as 
are  most  customary  for  the  several  types;  and  for  " normal  speed"  on 
the  part  of  the  subjects.  The  data  which  were  obtained  are,  therefore, 
representative  of  normal  performances  on  the  parts  of  the  readers  with 
these  three  types  of  materials.  Table  XXXIII  was  designed  to  facilitate 
the  comparison. 

TABLE  XXXIII 

COMPARATIVE  DATA  FROM  READINGS  OF  FIVE  PROBLEMS,  ORDINARY  PROSE,  AND 

ISOLATED  NUMERALS 
(Time  unit=  1/50  of  a  second) 


SUBJECTS 

AVERAGE 

FOR 

ALL 

SUBJECTS 

G 

B 

M 

w 

H 

Hb 

Average  number  of  pauses  per  line  on: 
The  five  problems  (first  reading) 

6.66 

4.71 

7-25 
7-5° 

8.41 

5-20 

7.25 

6.  20 

9.83 
6.66 

9.08 

8.08 
6.05 

The  ordinary  prose  selection  
Isolated  numerals 

Average  number  of  letters,  or  digits,  per  pause 
on: 
The  five  problems  (first  reading)  

6.61 
9.61 

2    28 

6.08 
6.03 
i  66 

i:S 

i  48 

6.08 
7.38 
i  62 

4-48 
6.75 
i  31 

4-85 

5.56 
7.69 
1.67 

11.88 
10.75 
23.80 

The  ordinary  prose  selection 

Average  duration  of  pauses  on: 
The  five  problems  (first  reading)  
The  ordinary  prose  selection 

11.28 

II.  IS 

30.57 

11-95 
10.77 
28.62 

10.93 
8.63 
19.85 

13-44 
12.13 
26.11 

12.  II 

11.25 

18.18 

11.58 

Isolated  numerals  

NOTE. — Satisfactory  data  on  the  reading  of  ordinary  prose  and  isolated  numerals  were  not  obtained 
from  Subject  Hb. 

The  average  line-length  of  the  five  problems  materials  was  93.33  mm.,  of  the  ordinary  prose  selection 
93.0  mm. 

The  conspicuous  feature  of  Table  XXXIII  is  the  marked  superiority 
in  the  speed  with  which  the  ordinary  prose  selection  was  read.  The 
greatest  differences  are  found  between  the  isolated  numerals  and  the 
prose  selection.  All  of  the  subjects  exhibit  these  differences,  both  in 


COMPARISONS  OF  RATES  OF  READING  97 

respect  to  average  number  of  digits  or  letters  read  per  pause  and  in 
respect  to  average  duration  of  pauses  as  well.  The  differences  indicating 
greater  speed  with  the  prose  are  in  large  proportion.  Such  differences 
in  speed  undoubtedly  reflect  the  great  and  obvious  differences  between 
materials  for  prose  and  for  isolated  numerals  in  respect  to  mechanical 
form,  context,  and  the  attitudes  which  they  inspire  in  the  minds  of 
readers.  Evidently  isolated  numerals  are  much  more  difficult  as  reading- 
material  than  ordinary  expository  prose. 

The  differences  in  rates  between  the  five  problems  and  the  ordinary 
prose  materials  are  decidedly  significant  although  not  as  great  in  propor- 
tion as  those  found  in  the  comparison  just  drawn.  Four  of  the  five 
subjects  read  the  ordinary  prose  at  higher  rates  than  they  had  used  with 
the  five  problems.  The  higher  speeds  with  the  prose  are  due  for  the 
most  part  to  fewer  pauses  per  line,  although  the  durations  of  the  pauses 
on  the  prose  are  somewhat  shorter  than  those  on  the  problems.  The 
obvious  interpretation  of  the  differences  is  found  in  the  greater  difficulty 
of  the  problems  as  reading-materials.  To  a  large  extent,  as  has  been 
shown  in  previous  sections  of  this  report,  the  greater  difficulty  is  due 
to  the  exactions  of  the  numerals  in  the  problems.  It  is  probable  also 
that  greater  exertions  were  undergone  by  the  subjects  in  their  efforts 
to  grasp  accurately  the  terms  of  arithmetical  problems  than  to  learn 
the  facts  contained  in  a  selection  of  ordinary,  expository  prose. 

3.     SUMMARY 

The  following  conclusions  may  be  drawn  from  the  discussion  of 
this  chapter:  (i)  The  subjects  of  the  present  investigation  read  at 
decidedly  higher  rates  than  the  subjects  of  Schmidt's  investigation. 
They  are,  therefore,  classified  as  decidedly  better  than  average  adults  in 
respect  to  speed  of  reading.  (2)  The  ordinary,  expository  prose  material 
was  read  at  significantly  higher  rates  than  either  the  arithmetical 
problems  or  the  isolated  numerals  materials.  The  conclusion  was 
drawn,  therefore,  that  arithmetical  problems  and  isolated  numerals  are 
decidedly  more  difficult  as  types  of  reading-materials  than  ordinary 
expository  prose. 


CHAPTER  XI 

PRACTICAL  APPLICATION  TO  CLASSROOM  TEACHING 
I.      THE   QUESTION   OF   READING   IN   ARITHMETIC 

Arithmetic  is  commonly  reputed  to  be  one  of  the  most  difficult  sub- 
jects in  the  curriculum  of  the  elementary  school.  The  meagerness  of 
the  results  which  are  obtained  at  the  cost  of  such  large  amounts  of  time 
as  are  devoted  to  the  study  of  the  subject  is  a  matter  of  continuous 
complaint.  The  changes  which  have  come  about  as  a  consequence  of 
efforts  to  improve  the  situation  have  resulted  for  the  most  part  in  a  new 
selection  of  subject-matter  for  textbooks  and  in  rearrangements  of  the 
sequence  of  topics.  The  problem  exercises  have  become  more  practical  in 
character  and  there  is  evidence  of  a  tendency  to  employ  in  problems  only 
such  classes  and  magnitudes  of  numerals  as  are  used  in  everyday  affairs. 

When  the  work  which  has  been  done  in  these  directions  is  duly 
recognized,  the  significant  fact  remains  that  some  of  the  most  important 
divisions  of  the  field  have  not  yet  been  occupied.  The  number  of 
scientific  studies  in  the  psychology  of  arithmetic  is  still  surprisingly 
small.  As  compared  with  reading,  arithmetic  has  been  seriously  neglected. 
Fortunately,  however,  much  of  the  information  which  has  been  made 
available  in  reports  on  reading  can  be  used  in  the  study  of  arithmetic. 
A  number  of  recent  reports  of  experiments  in  the  teaching  of  reading 
has  served  to  emphasize  the  existence  of  numerous  distinct  types  of 
reading  materials  each  of  which  calls  for  the  development  of  an  appro- 
priate type  of  reading  ability  on  the  part  of  the  student.  Arithmetical 
problems  and  isolated  numerals  as  well,  when  considered  as  reading 
situations,  represent  distinct  types  of  reading  materials.  The  special 
types  of  procedure  which  adult  subjects  used  in  reading  such  materials 
have  been  described  in  previous  chapters  of  this  report. 

Whatever  the  nature  of  the  materials,  however,  reading  is  essentially 
the  process  of  getting  meaning  from  the  printed  page.  When  the  subject- 
matter  is  difficult,  patient  and  systematic  search  for  the  thought  is 
necessary.  Instead  of  taking  pains  in  this  fashion,  children  frequently 
resort  to  mechanical  pronunciation  of  the  words  or  mere  scanning  of  the 
lines.1  In  such  cases  no  adequate  idea  of  the  meaning  is  obtained,  and 

1  Estaline  Wilson,  "Improving  the  Ability  to  Read  Arithmetic  Problems,"  Ele- 
mentary School  Journal,  Vol.  XXII,  No.  5  (January,  1922),  pp.  380-86. 

98 


PRACTICAL  APPLICATION  TO  CLASSROOM  TEACHING  99 

if  the  materials  read  be  arithmetical  problems,  a  correct  solution  is 
impossible.  It  is  scarcely  an  exaggeration  to  say  that  few  children, 
including  those  with  the  experience  of  the  upper  grades,  have  developed 
a  method  of  reading  problems  which  could  be  described  as  a  rapid  and 
skilful  attack  on  the  reading  situation. 

The  extent  to  which  the  appearance  of  numerals  along  with  words 
in  the  text  of  a  problem  is  responsible  for  this  condition  is  but  slightly 
understood.  Numerals  as  a  distinct  and  significant  object  of  study 
have  received  very  little  attention  in  the  literature  of  arithmetic.  Occa- 
sional reference  is  made  to  the  abstract  character  of  the  numerical  con- 
cept and  the  difficulty  of  teaching  children  the  meaning  of  the  symbols. 
Discussions  of  classroom  experience  also  are  reported  occasionally.  Of 
these  the  question:  What  magnitudes  of  numerals  should  be  used  when 
new  problems  are  introduced?  is  fairly  illustrative.  Numerals  in  the 
context  of  problems,  but  for  exceptions  of  this  character,  have  been 
almost  completely  neglected. 

2.      PRELIMINARY  ANALYTICAL   READING   OF   PROBLEMS 

The  presence  of  numerals  among  words,  however,  is  only  one  of 
the  features  which  distinguish  arithmetical  problems  as  a  specialized 
type  of  reading  materials.  The  existence  of  other  equally  complicating 
features  is  implied  when  teachers  express  the  opinion  that  it  is  more 
difficult  to  teach  the  interpretation  of  problems  than  the  mechanical 
skills  of  computation.  Despite  general  acceptance  of  this  opinion, 
scientific  studies  in  the  psychology  of  arithmetic  have  been  concerned 
almost  entirely  with  the  field  of  operations.  As  a  consequence,  exact 
knowledge  of  the  nature  of  the  processes  of  interpretation  is  lacking  for 
the  most  part. 

Efforts  have  been  made,  notwithstanding,  upon  the  basis  of  common 
observation  to  describe  the  reading  situation  which  a  problem  offers  and 
the  preliminary  thinking  about  it  which  is  necessary  before  the  reader  is 
prepared  for  the  work  of  computation.  The  most  essential  character- 
istic of  a  problem  is  the  fact  that  it  presents  a  series  of  conditions  which 
describe  a  certain  state  of  affairs.  Some  of  the  conditions  appear  in 
precise  quantity.  The  quantities  stand  in  definite  relationships  with 
each  other  and  are  stated  in  abstract  terms. 

Each  of  the  elements  of  this  complex  situation  must  be  compre- 
hended by  the  student  during  the  preliminary  reading.  He  must  draw  in 
his  imagination  an  accurate  picture  of  the  situation,  which  is  described, 
and  take  account  of  each  of  the  facts  of  relation.  Following  this,  comes 


100  HOW  NUMERALS  ARE  READ 

a  canvass  of  the  plans  for  solving  which  suggest  themselves,  and  the 
passing  of  judgment  on  the  appropriateness  of  each  plan  to  the  con- 
ditions of  the  problem.  The  reading  processes  which  are  carried  on  in 
this  manner  are  analytical  in  character  and  call  for  a  high  degree  of  skill 
and  patience,  as  well  as  for  a  certain  amount  of  practical  acquaintance 
with  the  facts  described.  It  is  doubtful  if  teachers  generally  have  acquired 
anything  like  an  adequate  appreciation  of  the  confusion  which  children 
feel  when  confronted  with  a  situation  to  be  studied  in  this  fashion. 

With  the  intention  of  facilitating  the  formation  of  working  habits 
which  will  bring  relief  to  this  confusion,  authors  of  teaching  manuals 
undertake  to  outline  the  steps  which  should  be  followed  in  reading  and 
solving  a  problem.  The  first  two  or  three  steps  usually  provide  for  the 
preliminary  analytical  reading.  The  following  directions,  which  appear 
in  a  recent  manual  as  the  first  three  of  five  steps,  will  serve  as  an  illustra- 
tion: "First,  the  pupil  must  read  the  problem  and  understand  what  it 
means.  Second,  he  must  state  in  his  own  words  what  the  problem  calls 

for Third,  he  must  find  the  material  that  the  problem  gives  to 

work  with."1 

The  reader  of  this  report  will  notice  immediately  that  no  suggestions 
are  given  concerning  the  proper  treatment  of  numerals.  As  far  as  the 
first  three  steps  are  concerned  it  would  seem  that  the  directions  were 
designed  for  use  in  connection  with  that  special  type  of  problem  only 
in  which  numerals  are  not  included.  The  author  appears  to  assume 
that  children  are  able  to  make  as  rapid  and  prompt  a  disposition  of  the 
numerals  as  the  adult  subjects  of  the  present  investigation.  It  is  clear 
that  a  rapid  and  thoughtful  reading  of  problems  containing  numerals  is 
not  practicable  under  the  directions  as  given  above,  unless  the  reader  is 
relieved  of  the  details  of  the  numerals  and  is  free  to  devote  his  entire 
attention  to  the  conditions  of  the  problem.  This  desirable  result,  as 
has  been  shown  in  a  previous  chapter  of  this  report,  can  be  obtained  by 
employing  the  partial  method  of  reading  numerals. 

3.      APPLICATION   OF  PARTIAL  READING 

The  importance  of  conducting  the  difficult  processes  of  analytical 
reading  in  the  most  direct  and  unhampered  manner  is  great  enough  to 
warrant  the  inclusion  with  the  directions  quoted  above  of  any  additional 
directions  that  may  be  necessary.  Undoubtedly  supplementary  direc- 
tions which  prescribed  partial  reading  of  the  numerals  would  greatly 
facilitate  the  preliminary  reading  of  problems.  Such  a  prescription 

1  Kendall  and  Mirick,  How  to  Teach  the  Fundamental  Subjects,  p.  169.  Boston: 
Houghton  Mifflin  Co. 


PRACTICAL  APPLICATION  TO  CLASSROOM  TEACHING          IOI 

would  also  serve  to  emphasize  the  significance  of  partial  reading, 
and  the  extensive  possibilities  of  improvement  which  lie  in  its  use  could 
begin  to  be  realized.  At  this  point,  the  reader  should  be  reminded  of 
the  fact  that  the  problems  which  were  employed  in  the  present  investiga- 
tion were  both  brief  and  simple.  Since  the  partial  method  facilitated 
the  reading  of  such  problems,  its  value  for  the  larger  and  more  involved 
problems  which  are  found  in  the  ordinary  textbook  would  be  corre- 
spondingly greater.  And  likewise,  a  relief  measure  which  was  desirable 
for  adult  graduate  students  should  prove  all  the  more  helpful  to  children 
in  the  elementary  school. 

An  effort  has  been  made,  therefore,  to  prepare  recommendations  con- 
cerning the  reading  of  problems  which  should  be  considered  as  additional 
to  the  directions  that  are  quoted  above.  The  recommendations  are  as 
follows: 

1.  Pupils  should  be  taught  to  distinguish  between  the  first  reading 
and  the  re-reading  phases  in  their  attack  on  problems. 

2.  They  should  learn  to  consider  numerals  and  the  accompanying 
descriptive  conditions  as  different  elements  of  a  problem  and  separable 
for  reading  purposes. 

3.  During  the  first  reading,  they  should  devote  their  entire  attention 
to  the  conditions  of  the  problem. 

4.  At  the  same  time  skill  should  be  developed  in  partial  reading  of 
numerals. 

5.  While  this  skill  is  being  acquired,  pupils  should  be  apprised  of  the 
essential  similarity  between  the  conditions  of  the  problem  and  such 
details  of  the  numerals  as  are  perceived  by  partial  reading. 

Although  the  preceding  recommendations  were  derived  for  the  most 
part  from  the  findings  of  this  report,  their  validity  does  not  rest  on  this 
basis  alone.  They  derive  additional  support  by  comparison  with  the 
findings  of  other  investigations  in  the  field  of  reading.  Gray,  in  sum- 
marizing the  principles  of  method  which  were  deduced  from  recent 
studies  of  reading,  states  that  emphasis  of  the  elements  on  which  mean- 
ing depends,  improves  comprehension.1  A  closely  related  principle  is 
stated  by  Freeman:  "Rate  of  reading  is  increased  by  attending  to  the 
meaning  as  distinguished  from  the  mechanics."2  In  the  case  of  arith- 
metical problems,  the  elements  referred  to  by  Gray,  obviously,  are  the 

1  W.  S.  Gray,  "Principles  of  Method  in  Teaching  Reading,  as  Derived  from 
Scientific  Investigation,"  Eighteenth  Yearbook  of  the  National  Society  for  the  Study  of 
Education,  Part  II,  p.  42. 

2  F.  N.  Freeman,  The  Psychology  of  the  Common  Branches,  p.  93.    Hough  ton 
Mifflin  Co. 


102  HOW  NUMERALS  ARE  READ 

conditions  of  the  problem  and  such  details  of  the' numerals  as  identity  of 
the  first  digit  and  the  number  of  digits.  Attention  to  these  items  and 
to  these  alone  is  obtained  by  the  use  of  partial  reading  as  recommended 
above.  The  remaining  details  of  the  numerals  are  of  the  nature  of 
"  mechanics, "  as  described  by  Freeman,  and  when  the  partial  method  of 
reading  is  used,  the  attention  of  the  student  is  relieved  of  the  "  mechanics  " 
and  is  free  to  search  out  the  meaning. 

Further  comparisons  with  the  results  of  other  investigations  empha- 
size the  fact  that  the  use  of  partial  reading  as  recommended  above  repre- 
sents a  more  progressive  type  of  reading.  Progress  in  reading,  according 
to  Freeman,  consists  in  a  decrease  of  the  number  of  pauses  of  the  eye 
and  an  increase  in  the  scope  of  recognition  at  each  pause.1  Gray  con- 
cludes that  "regular  rhythmical  movements  of  the  eyes  are  prerequisite 
to  rapid  silent  reading."2  When  partial  reading  is  used,  the  numerals  of 
a  problem  are  not  read  in  detail  and  it  has  been  shown  that  fewer  pauses 
are  required  to  get  the  meaning  from  the  printed  line.  As  a  conse- 
quence, the  average  amount  of  material  perceived  at  a  pause  is  increased. 
It  is  clear,  therefore,  that  both  of  the  conditions  which  Freeman  describes 
as  representative  of  progress  in  reading  are  encouraged  by  the  method 
of  partial  reading.  At  the  same  time,  by  employment  of  this  method, 
the  eye  is  relieved  of  the  most  severe  exactions  of  the  numerals  and  does 
not  suffer  the  delay  in  movement  which,  owing  to  the  nature  of  numerals 
as  reading  materials,  is  unavoidable  when  the  numerals  are  read  in  detail. 
By  virtue  of  this  relief,  the  .eye  is  able  to  approximate  more  nearly 
the  rhythm  of  movement  which  is  customary  for  lines  of  words,  and 
which  is  "prerequisite  to  rapid  silent  reading." 

4.      APPLICATION   OF   RE-READING 

The  recommendations  which  appear  above  are  concerned  exclusively 
with  the  first  reading  of  the  problem.  When  the  first  reading  is  com- 
pleted ordinarily  the  pupil  is  ready  for  the  re-reading.  It  is  possible, 
as  was  shown  in  chapter  viii,  to  omit  the  work  of  re-reading  by  employing 
the  method  of  mental  computation  while  reading  the  problem  as  printed. 
The  latter  is  a  more  economical  procedure  than  the  alternative  method 
of  computation  from  copied  figures.  So  rapid  a  procedure  as  direct 
computation,  however,  is  practicable  only  with  easy  problems  and  for 
pupils  of  unusual  arithmetical  ability.  Probably  the  great  majority  of 
pupils  solve  the  largest  proportion  of  their  problems  with  the  aid  of 
pencil  and  paper. 

JF.  N.  Freeman,  op.  cit.,  p.  83.  2  W.  S.  Gray,  op.  cit.,  p.  39. 


PRACTICAL  APPLICATION  TO  CLASSROOM  TEACHING          163 

For  all  such  pupils  the  work  of  re-reading  and  copying  the  numerals 
is  an  essential  step  in  the  process  of  solving.  Extensive  use  of  a  method, 
however,  does  not  guarantee  successful  accomplishment.  By  experi- 
ence, and  by  careful  investigation  as  well,  teachers  have  learned  that 
pupils  do  not  copy  numerals  accurately.  The  situation,  as  it  exists,  is 
too  serious  to  be  neglected.  Intelligent  and  vigorous  efforts  should  be 
made  to  eliminate  errors  of  this  kind  completely.  Substantial  improve- 
ment undoubtedly  could  be  effected,  by  relieving  the  pupil's  mind  of  every 
avoidable  distraction  during  the  copying.  Setting  apart  a  definite  place 
for  the  step  of  copying,  in  the  total  procedure  of  solving,  would  consti- 
tute an  important  move  in  this  direction.  For  this  reason,  the  teacher's 
plan  of  instruction  for  arithmetical  problems  should  include  a  special 
period  of  drill  in  re-reading  and  copying  numerals. 

When  the  plan  of  instruction  has  been  amended,  in  accordance  with 
this  suggestion,  it  is  important  that  the  most  efficient  methods  of  reading 
the  numerals  be  selected  for  use  during  the  drill  period.  The  methods 
of  re-reading  which  were  employed  by  the  adult  subjects  of  the  present 
investigation  are  suggestive  of  the  type  of  drill  which  should  be  given. 
For  this  reason  the  conclusions  which  were  derived  in  chapter  v  were 
taken  as  the  basis  of  recommendations  concerning  re-reading  which  are 
submitted  as  follows:  (i)  the  numerals  of  any  digit  length  should  be 
read  according  to  their  dominant  main-group  pattern;  and  (2)  the 
simplest  possible  numerical  language  should  be  used. 

The  effect  of  these  recommendations  and  the  advantages  that  lie 
in  their  adoption  may  be  illustrated  with  the  numerals  56,283  and  497. 
When  the  numeral  56,283  is  treated  as  directed,  it  is  read  as  follows: 
five  six,  two  eight  three.  In  this  way  the  main-group  pattern  for  five- 
digit  numerals,  which  calls  for  two  groups  of  two  and  three  digits  respec- 
tively, is  followed.  No  words  are  used  except  such  as  are  required  to 
name  the  digits  in  order  of  succession.  In  the  case  of  the  numeral  497 
the  proper  pronunciation  is  four,  nine  seven.  In  this  instance  the 
dominant  main-group  pattern  for  three-digit  numerals  is  followed  and 
no  superfluous  language  is  included. 

In  the  reading  of  both  numerals,  advantage  is  taken  of  the  strong 
natural  tendency  of  the  mind  to  arrange  the  members  of  a  series  of 
stimuli  in  groups.  With  the  verbal  description  as  brief  as  possible,  no 
unnecessary  waste  of  energy  is  incurred  in  pronunciation.  In  addition 
to  the  value  of  economy,  an  important  reduction  is  effected  in  oppor- 
tunity for  errors  by  reading  the  numerals  exactly  as  they  are  printed 
and  precisely  in  the  form  in  which  they  are  to  be  copied.  The  feasi- 


104  HOW  NUMERALS  ARE  READ 

bility  of  the  recommendations  is  beyond  doubt  since  they  are  observed 
regularly  in  practice  by  various  vocational  groups,  such  as  telephone 
operators  and  bookkeepers,  who  use  numerals  extensively  in  their 
daily  work. 

Nor  are  the  directions  as  given  applicable  only  to  oral  reading.  It 
is  a  matter  of  common  knowledge  among  students  of  reading  that  a 
very  close  connection  exists  between  the  inner  speech  of  silent  reading 
and  the  behavior  of  oral  reading.  Numerous  elements  of  procedure  are 
common  to  both.  So  far  as  the  foregoing  recommendations  are  con- 
cerned, there  appears  no  necessity  for  drawing  a  distinction  between  oral 
and  silent  reading. 

5.      MISCELLANEOUS   APPLICATIONS 

Although  the  findings  of  the  present  investigation  are  not  concerned 
extensively  with  the  processes  of  computation,  important  data  were 
presented  in  chapter  viii  concerning  "computation  direct  from  the 
problem  card."  The  records  show  that  this  method  is  a  very  direct 
road  to  the  answer  and  its  use  undoubtedly  reduces  the  number  of  oppor- 
tunities for  error.  The  great  rapidity  with  which  the  abler  students 
can  solve  suitable  problems  in  this  manner  would  tend  to  increase  the 
pupils'  interest  and  concentration.  For  these  reasons  it  is  recommended 
that  with  abler  students  the  process  of  direct  computation  be  more 
extensively  employed  in  rapid  drill  with  problems. 

The  question  will  be  raised  as  to  the  grade  at  which  the  methods  of 
partial  reading  and  re-reading  should  be  introduced.  The  findings  of 
the  present  investigation  do  not  bear  directly  on  this  question.  Nor 
can  other  than  provisional  conclusions  be  drawn  until  the  reading  of 
problems  by  children  in  the  various  grades  has  been  studied.  Signifi- 
cant inferences,  however,  can  be  drawn  in  the  light  of  certain  principles 
of  method  in  teaching  reading  which  were  derived  from  scientific  study 
and  which  are  now  generally  accepted. 

Partial  reading  and  re-reading  are  methods  of  skilful  and  rapid 
silent  reading.  They  represent  a  degree  of  achievement  which  is  more 
advanced  than  mere  ability  to  recognize  the  words.  It  is  reasonably 
certain,  therefore,  that  children  are  not  prepared  to  read  in  this  fashion 
before  the  teaching  emphasis  has  been  shifted  from  oral  to  silent  reading, 
which,  as  is  now  generally  known,  should  be  done  between  the  second 
and  fourth  grades.  Beginning  with  the  latter  grade,  progress  in  reading 
consists  in  large  part  in  ability  to  master  increasingly  difficult  materials. 
Arithmetical  problems  with  several  conditions  and  with  longer  numerals 


PRACTICAL  APPLICATION  TO  CLASSROOM  TEACHING          105 

constitute  such  materials  and  it  is  this  type  of  problem  which  is  attacked 
to  advantage  by  the  use  of  partial  reading.  In  the  nature  of  the  case, 
partial  reading  and  re-reading  are  highly  specialized  types  of  procedure. 
It  is  during  the  fourth,  fifth,  and  sixth  grades  that  pupils  should  be 
trained  to  use  different  types  of  reading  ability.  In  view  of  this  con- 
sideration, and  of  such  others  as  are  named  above,  it  appears  that  the 
fourth  grade  is  the  appropriate  time  for  the  introduction  of  the  new 
methods. 

The  discussion  of  applications  ought  not  to  be  concluded  without 
pointing  out  the  fact  that  a  body  of  experimental  data  in  many  instances 
is  valuable  for  other  purposes  than  those  which  were  originally  respon- 
sible for  the  investigation.  Diagnosis  of  individual  difficulties  is  an 
increasingly  important  feature  of  instruction  in  modern  school  systems. 
For  such  work,  well- trained  teachers  and  supervisors  rely  to  as  great 
an  extent  as  is  possible  upon  the  materials  which  are  available  in  scientific 
reports.  No  other  body  of  material  affords  an  equally  detailed  and 
illuminating  description  of  the  intellectual  processes  which  are  carried 
on  in  the  ordinary  work  of  the  school.  The  description  of  first  reading 
and  re-reading  and  of  certain  computational  processes,  which  is  avail- 
able in  the  present  report,  will  prove  useful  for  diagnosis  of  difficulties 
with  arithmetical  problems.  Significant  but  less  detailed  descriptions 
of  the  range  of  correct  recall  of  numerals,  the  grouping  of  digits,  numeri- 
cal-language patterns,  effect  of  punctuation,  and  of  such  other  items  as 
a  perusal  of  the  table  of  contents  will  disclose,  are  also  available  for  the 
same  purpose. 


INDEX 


Articulation    of  numerals,  24,  38,  61 

"Base  of  operations":  one  numeral  used 
as,  74-76 

Combinations  of  digits,  57,  67 
Complex  three-digit  groups,  27 
Computation   from   copied   figures:    de- 
scribed, 72;   wide  use  of,  102 
Copying  numerals,  103 

Dearborn,  W.  F.,  57,  78 
Diagnosis  of  pupil  difficulties,  105 
Digit  groups:  code  used  in  reporting,  25; 
description    of,    25;     and    habits    of 
subjects,  27;  influence  of  punctuation 
on,  31;    of  longer  and  shorter  numer- 
als, 27-29;  types  of  three-digit  groups, 
27 

Direct  computation:  contrasted  with 
computation  from  copied  figures,  72; 
defined,  9;  pedagogical  significance  of, 
73;  recommendation  concerning,  104; 
when  practicable,  102;  where  found,  73 
Directions  for  reading  problems:  five 
recommendations,  101;  by  Kendall 
and  Mirick,  100 

Essential  characteristics  of  a  problem,  99 

Familiar  numerals:  explanation  of,  63; 
main-group  patterns  of,  28;  partial 
reading  of,  5;  and  range  of  recall,  15; 
special  treatment  of,  85 

Familiarity  of  form  in  numerals,  90 

Final  pauses  on  numerals,  85 

First  digits  of  numerals:  correctly  re- 
called, 15,  67;  pauses  located  on,  86 

First  numeral  in  line,  79 

First  numerals:  and  partial  reading,  15; 
and  range  of  recall,  15 

First  reading:  defined,  3;  purpose  of ,  1 7 ; 
recommendations  concerning,  101; 
time  required,  23 

Freeman,  F.  N.,  101,  102 

Gilliland,  A.  R.,  35 
Gray,  C.  T.,  35,  78 


Gray,  W.  S.,  101,  102 
Guiding  pauses,  79,  86,  91 

Initial  pause  on  a  line  of  numerals,  79,  85 

Interest  in  problems,  61 

Introspective  observations,  i,  39,  59,  69 

Instructions  to  subjects  for:  eye-move- 
ment studies,  38;  first  preliminary 
study,  3;  fourth  preliminary  study,  24; 
second  preliminary  study,  13;  third 
preliminary  study,  19 

Isolated  numerals:  average  number  of 
pauses  on,  90;  compared  with  problem 
numerals,  60;  of  eye-movement  studies, 
37;  of  fourth  preliminary  study,  24; 
nature  of,  as  reading  materials,  96; 
reading  of  the  special,  90 

James,  William,  94 
Judd,  C.  H.,  38,  94 

Kendall  and  Mirick,  100 

Last  digits  of  numerals,  86 

Locating  pauses,  74-76 

Longer  numerals:  influence  of  punctua- 
tion on  grouping  of  digits,  31,  32; 
number  of  pauses  used  in  reading,  90; 
partial  and  whole  readings  of,  59,  61, 
62;  partial  reading  of,  5;  range  of 
recall  of,  15;  re-reading  for  copying, 
23;  used  as  "base  of  operations,"  76 

Main-group  patterns:    definition  of,  27; 

recommendations   concerning   use   of, 

103;  of  various  sizes  of  numerals,  28 
Many-short-pauses    method    of    reading 

numerals,  91 
Methods  of  computation:   two  types  of, 

72 
Methods  of  reading  numerals,  two,  91 

Numerals:  articulation  of,  24;  classes 
studied,  i;  non-punctuated,  31;  as 
peculiar  reading  materials,  22;  previ- 
ous investigations  of,  i ;  snorter  range 
of  perception  of,  56;  the  special,  90; 
total  reading  time  of,  86;  used  in  eye- 
movement  studies,  86;  used  in  first 


107 


io8 


NOW  NUMERALS  ARE  READ 


preliminary  study,  2;  used  in  fourth 
preliminary  study,  24 
Numerical  language:  code  used  in  re- 
porting, 25;  and  habits  of  subjects,  31; 
persistence  of  patterns,  33;  and 
punctuation,  33 ;  recommendations 
concerning  use  of,  103;  variations  in, 
29-31 

Ordinary   prose    selection,    38;     rate    of 
reading,  96 

Partial  first  readings,  59 
Partial  readers:  attitude  of,  66;  classi- 
fication of,  7,  64;  prose  reading  rate 
of,  65;  range  of  perception,  56;  and 
range  of  recall,  17;  reading  of  words 
by,  65;  re-reading  of,  9,  71;  use  fewer 
pauses,  66 

Partial  reading:  of  the  date,  1918,  5; 
description  of,  3,  67;  development  of, 
66;  and  direct  computation,  73;  and 
the  essential  elements  of  a  problem, 
101;  of  familiar  numerals,  5;  of  first 
numeral,  6;  frequency  of,  4;  of 
individual  subjects,  6;  introduce  at 
fourth  grade,  104;  of  larger  and  shorter 
numerals,  5;  number  of  digits  included 
in  a  pause,  63;  and  progress  in  reading, 
102;  of  several  numerals  in  one 
problem,  5;  value  of,  4,  64,  100 
Pauses  on  numerals :  average  duration  of, 
57,  86,  90;  duration  of,  on  isolated 
numerals,  79;  final  pauses,  85;  greater 
duration  of  "base  of  operations" 
pauses,  76;  guiding  pauses,  79,  86,  91; 
initial  pause  on  a  line  of  numerals,  79, 
85;  locating  pauses,  74-76;  pairs  of, 
85;  range  of  perception  of,  57;  recogni- 
tion pauses,  76;  recording  pauses, 
74-76;  regressive  pauses,  58,  79;  two 
types  of,  85 
Photographic  apparatus:  description  of, 

35;  films,  39;  use  of,  i 
Practice  numerals,  25 
Practice  problems,  13,  20,  39 
Problem-solving  attitude,  13,  38 
Problems  as  reading  materials,  95~97,  99 
Problems  used  in:  eye-movement  studies, 
36,    73;     first    preliminary    study,  ^2; 
second  preliminary  study,   12;    third 
preliminary  study,  19,  36 
Problems  without  numerals,  100 
Progress  in  reading  by  partial  reading, 


Punctuation:  and  digit  groups,  31;  and 
main-group  patterns,  32;  and  numer- 
ical language,  33;  value  of,  32 

Range  of  correct  recall :  classifications  of, 
13;  complete,  15;  extent  of,  14;  of 
first  numerals,  15;  most  frequent,  14; 
of  shorter  numerals,  15;  according  to 
subjects,  16;  unclassified  items,  17 

Range  of  perception,  56;  and  develop- 
ment of  partial  reading,  66,  67;  of 
digits,  63,  79,  92;  of  Subject  G,  92 

Rate  of  reading:  different  materials,  96; 
explanation  of  higher  rates,  97;  Free- 
man on,  101;  prose  by  partial  and 
whole  readers,  65;  of  Schmidt's  sub- 
jects, 95;  of  subjects  of  present 
investigation,  95 

Reading  materials:  different  types  of,  98; 
isolated  numerals  as,  96;  ordinary 
prose  as,  96;  problems  as,  95-97 

Recognition  pauses,  76 

Recommendations  concerning :  reading 
of  problems,  101;  re-reading  and 
copying  numerals,  103 

Recording  pauses,  74-76 

Regressive  pauses:  in  detailed  reading, 
85;  explanation  of,  58;  initial,  79,  85; 
on  words  and  numerals,  58 

Regularity  of  form  in  numerals,  90 

Re-reading:  dependent  on  habits  of 
subjects,  71;  on  length  of  numerals,  8; 
distinguished  from  first  reading,  3; 
following  whole  reading,  5,  9;  and 
mental  computation,  9,  10;  numerals, 
objects  of,  22;  percentages  of,  8; 
purpose  of,  3,  71;  two  types  of,  69 

Re-reading  for  copying:  description  of, 
22,  71;  introduce  at  fourth  grade,  104; 
recommendations  concerning,  103; 
several  numerals  in  one  problem,  23; 
shorter  and  longer  numerals,  23 

Re-reading  time,  23 

Rythmical  movements  of  the  eye  and 
partial  reading,  102 

Schmidt,  W.  A.,  94,  95 
Several     numerals     in     one     problem: 
'  partial  reading  of,  5,  6,  63;    re-reading 

of,  23 
Shorter    numerals:     partial    and    whole 

reading  of,  61;    partial  reading  of,  5; 

range  of  recall  of,  15;    re-reading  for 

copying,  23 


INDEX 


109 


Simple  re-reading,  69-71 
Simple  three-digit  groups,  27 
Special  numerals,  90 

Subjects  who  served  in:  eye-movement 
studies,  39;  first  preliminary  study,  2; 
fourth  preliminary  study,  24;  second 
preliminary  study,  12;  third  pre- 
liminary study,  19 

Total  reading  time  of  numerals,  86; 
shortest,  91 

Visual  memory,  10 


Whole  first  readings,  59;  of  isolated 
numerals,  61 

Whole  readers:  attitude  of,  66;  classi- 
fication of,  7,  64;  prose  reading,  rate 
of,  65;  range  of  perception,  56;  range 
of  recall,  17;  reading  of  words  by,  65; 
re-reading  of,  9,  71 

Whole  reading:  description  of,  4;  and 
direct  computation,  73;  explanation 
of,  10;  relative  value  of,  64;  used  by 
beginners,  66 

Wilson,  Estaline,  98 

Words:  range  of  perception  of,  56; 
reading  of,  by  partial  readers,  65 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


